I am not asking about any resolution he might have made in the feverish April of 1945.

Do we have any information how Hitler imagined the government of his Reich after he would have died from old age? Did he wish to appoint a specific successor? Produce an heir? Or would they transition to some form of rule other than one-person dictatorship?

Note: My question is not duplicate of this. It asks about who would have succeeded Hitler had he died before 1945, and Göring is definitely not the answer to my question, as he was only four years Hitlers junior.

The official, factual side

As unsatisfactory as that might sound @JMS is rightly focussing on first Göring and then Dönitz/Goebbels, although the later declined to survive so that Schwerin von Krosigk was stepped up.

But that is really it, as this is most typical of a dictatorship that is based on the love of the people for that very person. This is called charismatic leadership or authority and the theory for it was already detailed and analysed by Max Weber years before Hitler was given power.

In these conditions designating a clear successor is inherently dangerous as it undermines the messianic character and therefore power of the dictator. Charismatic leadership means abolishment of institutions. One of those would be the regulations for naming a successor publicly.

Hitler himself found that to be the case when in 1945 Göring assumed this role in earnest and asked quite politely and carefully whether the old, hastily drafted and naturally seldom discussed "succession law" would come into effect, either by direct confirmation or if he hadn't heard from the encircled Führer.

All at once, there was a commotion in the corridor. Bormann hurried in with a telegram for Hitler. It was from Göring. The report of the momentous meeting the previous day, which Koller had personally flown to Berchtesgaden to deliver verbally, had placed the Reich Marshal in a quandary. Koller had helped persuade a hesitant Göring that, through his actions, Hitler had in effect given up the leadership of state and Wehrmacht. As a consequence, the edict of 29 June 1941, nominating Göring as his successor in the event of his incapacity to act, ought to come into force. Göring was still unsure. He could not be certain that Hitler had not changed his mind; and he worried about the influence of his arch-enemy, Bormann. Eventually, Koller suggested sending a telegram. Göring agreed. Koller, advised by Lammers, drafted its careful wording, cautiously stipulating that, had Göring not heard by ten o’clock that evening, he would presume that the terms of the succession law would come into operation, and that he would take over the entire leadership of the Reich.
_{Ian Kershaw: "Hitler", Penguin Books: London, 2013.}

This underdeveloped care for the future is already seen very clearly in how Alexander the Great planned his succession. While he himself came to power through traditional monarchical inheritance of the throne, at the latest after he took the Persian Crown his leadership and authority was transformed from the traditional type into the charismatic. No-one could match him. Only on his death-bed he replied when asked about his succession that "the best shall do it". As we know, a nice recipe for disaster.

Not only do charismatic leaders avoid the issue until the very end, historians seem equally largely disinterested in analysing this phenomenon.

While the often colorful story of the rise of dictators and their movements has been given a good deal of attention within and without their regimes, much less interest has been shown in the problem of their continuation after a demise of the "leader." It is true that in most cases the problem of succession did not arise because dictatorial rule was terminated through the premature, and more or less abrupt and violent, end of the regime as such. But even prior to such event, little public attention is given to the problem within dictatorships. The mystique of the leader considers him as unique, and raising the problem of his demise, even that of his natural death, would put his uniqueness to doubt and pull his regime down to the level of any ordinary type of government. In totalitarian dictatorships the problem is hardly ever discussed. As far as one knows there is no public mention inside the Soviet Union of the question of what will happen after Stalin's death. There was no such discussion in Nazi Germany until Hitler's somewhat casual remark, at the outbreak of the war, as to what should happen "if anything should happen to me in the struggle."
_{John H. Herz: "The Problem of Successorship in Dictatorial Régimes; A Study in Comparative Law and Institutions", The Journal of Politics, Vol. 14, No. 1 (Feb., 1952), pp. 19-40. (jstor)}

So the short answer to

Q Had things gone his way, how did Hitler imagine his succession?

is: he just failed to do that.

(And admitted that directly, cf. Hugh Redwald Trevor-Roper: "The Last Days of Hitler", Macmillan: New York, 1947), pp. 91 ff., 129.)

The "imagined" side

While at the end most nazis would expect Himmler to take the baton, once Göring was out of the picture, the question here seems to focus on Hitler himself and his ideas. There are no official speeches or documents indicating anything in that direction, other than these plans for Göring etc.

If one is willing to accept the following source as genuine and reliable, and is willing to engage in some interpretation into the direction "what he might have imagined, then a few surprises might come along:

Without a solid political basis, it's not possible either to settle a question of succession or to guarantee the normal administration of the State. From this point of view, the Rumanians are in a state of inferiority in relation to the Hungarians. The Hungarian State has the advantages of a parliament. For us, such a thing would be intolerable; but theirs is one whose executive power is, in practice, independent.
_{(18th January 1942, evening)}

Setting the best man at the head of the State—that's the most difficult problem in the world to solve.[…]

In a republic that sets at its head a chief elected for life, there's the risk that he will pursue a policy of personal self-interest.
In a republic where the Chief of State changes every five or ten years, the stability of the government is never assured, and the execution of long-term plans, exceeding the duration of a lifetime, is thereby compromised.

If one sets at the head of the State an old man who has withdrawn from all worldly considerations, he is only a puppet, and inevitably it's other men who rule in his name.

Thinking over all that, I've arrived at the following conclusions:

1. The chances of not setting a complete idiot at the head of the State are better under the system of free elections than in the opposite case. The giants who were the elected German Emperors are the best proof of this. There was not one of them of whom it can truly be said that he was an imbecile. In the hereditary monarchies, on the other hand, there were at least eight kings out of ten who, if they'd been ordinary citizens, would not have been capable of successfully running a grocery.


2. In choosing a Chief of State, one must call upon a personality who, as far as human beings can judge, guarantees a certain stability in the exercise of power for a longish while. This is a necessary condition, not only so that public affairs can be successfully administered, but in order to make possible the realisation of great projects.


3. Care must be taken that the Chief of State will not succumb to the influence of the plutocracy, and cannot be forced to certain decisions by any pressure of that sort. That's why it's important that he should be supported by a political organisation whose strength has its roots in the people, and which can have the upper hand over private interests.

In the course of history, two constitutions have proved themselves:

(a) The papacy, despite numerous crises—the gravest of which, as it happens, were settled by German emperors—and although it is based on a literally crazy doctrine. But as an organisation on the material level, the Church is a magnificent edifice.

(b) The constitution of Venice, which, thanks to the organisation of its Government, enabled a little city-republic to rule the whole eastern Mediterranean. The constitution of Venice proved itself effective as long as the Venetian Republic endured — that is to say, for nine hundred and sixty years.

As regards the government of Germany, I've come to the following conclusions:

1. The Reich must be a republic, having at its head an elected chief who shall be endowed with an absolute authority.


2. An agency representing the people must, nevertheless, exist by way of corrective. Its role is to support the Chief of State, but it must be able to intervene in case of need.


3. The task of choosing the Chief shall be entrusted, not to the people's assembly, but to a Senate. It is, however, important that the powers of the Senate shall be limited. Its composition must not be permanent. Moreover, its members shall be appointed with reference to their occupation and not individuals. These Senators must, by their training, be steeped in the idea that power may in no case be delegated to a weakling, and that the elected Fuehrer must always be the best man.


4. The election of the Chief must not take place in public, but in camera. On the occasion of the election of a pope, the people does not know what is happening behind the scenes. A case is reported in which the cardinals exchanged blows. Since then, the cardinals have been deprived of all contact with the outside world, for the duration of the conclave! This is a principle that is also to be observed for the election of the Fuehrer: all conversation between (? with) the electors will be forbidden throughout operations.


5. The Party, the Army and the body of officials must take an oath of allegiance to the new Chief within the three hours following the election.


6. The most rigorous separation between the legislative and executive organs of the State must be the supreme law for the new Chief. Just as, in the Party, the SA and the SS are merely the sword to which is entrusted the carrying-out of the decisions taken by the competent organs, in the same way the executive agents of the State are not to concern themselves with politics. They must confine themselves exclusively to ensuring the application of laws issued by the legislative power, making appeal to the sword, in case of need. Although a State founded on such principles can lay no claim to eternity, it might last for eight to nine centuries. The thousand-year-old organisation of the Church is a proof of this—and yet this entire organisation is founded on nonsense. What I have said should a fortiori be true of an organisation founded on reason.

_{(3 March 1942, at dinner)}

As regards the Head of the State, should anything happen to me, it would be as unsound to elect my successor by public vote as it would for, say, the Pope to be elected by suffrage among the faithful, or the Doge of Venice by the vote of the whole population of the city. If the mass of the people were invited to take part in such a vote, the whole thing would degenerate into a propaganda battle, and the propaganda for or against any candidate would tear the people asunder.

If the choice is left to a small body —a senate, for instance— and marked differences of opinion should arise in it, I don't think it would matter very much, provided that no hint of these differences was allowed to become public. But once the votes have been cast, then he who receives the majority becomes automatically and forthwith the supreme head of the state. If it is further arranged that the oath of allegiance to the new Head can be administered to the Wehrmacht, the Party and all the appropriate officials within three hours of the result of the election, then maintenance of public law and order can be regarded as assured.

_{(24th June 1942, at dinner)}

_{Norman Cameron and R.H. Stevens (translators), Hugh Redwald Trevor-Roper: "Hitler's Table-Talk 1941–1944. His Private Conversations", Enigma Books: New York, 2000.}

Note that this source is entertaining and interesting, but very probably not overly reliable (and the quality of translation leaving much to be desired)! Even if exactly spoken in that way, by then the person would ramble on quite a bit and contradict himself quite often. After the two instances listed above, according to those collections assembled by a nazi, he never again talked about the future, other than in terms of architecture (not as in administration or state construction, but brick and mortar) or grandiose ideas of conquering the four corners of the world. Nothing of these 'plans' – or better preliminary ideas – for a future leader was set in motion in any way.

Question:

Had things gone his way, how did Hitler imagine his succession?

Background:

Health of Adolf Hitler

Hitler's physical health has long been the topic of speculation.
Physically Hitler suffered from tremors and irregular heartbeat during the last years of his life. Hitler's personal doctor Dr. Theodor Morell diagnosed him with tertiary (late stage) syphilis in early 1945 in a joint report to SS head Heinrich Himmler. Tertiary means he suffered from it for many years of his life.

In 1942 Hitler suffered from severe headaches, dizziness and insomnia.
Dr. Felix Kersten who was consulted at that time, told his biographer Joseph Kessel that he saw a top-secret report which detailed how Hitler had contracted syphilis in his youth and how symptoms began to reassured in 1937.
That by 1942 he was suffering from progressive syphilitic paralysis(Tabes dorsalis)

Other health issues which have been attributed to Hitler.

• Parkinson's disease


• Huntington's disease


• Hypochondria


• Chronic Drug Use

Prescribed 90 medications during the war years by Dr. Theodor Morell, Hitler took many pills each day for chronic stomach problems and other ailments. He regularly consumed methamphetamine, barbiturates, opiates, and cocaine, as well as potassium bromide and atropa belladonna (the latter in the form of Doktor Koster's Antigaspills).

Answer:

Hitler envisioned the third Reich lasting for 1000 years. With his poor health and hypochondria he had made plans for secession in the event of his demise. In Albert Speer's book Inside the Third Reich: Recollections he mentioned that Hitler wanted to retire to Austria after winning WWII. Speer makes no mention of whom Hitler envisioned would succeed him in a Reich not besieged by it's enemies.

Operation Valkyrie was a Nazi plan for continuity of government operations in the case of an emergency, such as Hitler's death. It called for a modified chain of command and orders of operation to defend the nation in such an event. This standing order was used July 20th, 1944, in a coup attempt after a bomb went off in Hitler's bunker, set by conspirators. German General Friedrich Olbricht, Major General Henning von Tresckow, and Colonel Claus von Stauffenberg modified this standing plan in their failed attempt to take control of German cities, disarm the SS, and arrest the Nazi leadership after their attempt on Hitlers life. The plan ultimately failed when Hitler who survived the attempted assassination was able to get word out that he survived, negating Operation Valkyrie's standing orders and cutting the conspirators off from the German Military which were being used without their knowledge to further the coupe.

Hermann Göring was the second in command to Adolf Hitler throughout WWII and was Hitler's designated successor. Hitler made this designation publicly first in a speech on the first day of the war in 1939, and them formally in a decree in 1941.

I know you are not interested in his late 1945 degrees but I wanted to included them as they are pertinent. Late April 1945 when Hitler was cut off and isolated in his bunker in Berlin; Göring sent a telegram asking if leadership now fell to him and that if he didn't hear back from Hitler by a certain time, he Göring, would assume command. Hitler infuriated stripped Göring of all titles and responsibilities, including eliminating him from the line of succession.

Here is an English translation of Hitler's final will in which Hitler references both the speech and decree leaving leadership to Hermann Göring and announces he is repealing both.

Hitler's Last will and Testament

The day before Hitler died April 29th, 1945 he appointed the commander of the Navy, Grand Admiral Karl Dönitz to President of Germany - Reichpräsident and Joseph Goebbels as Chancellor of Germany - Reichskanzler. Hitler had combined these two offices under himself as Fuehrer in 1934 when Paul Von Hindenburg died.

Source:

• 
  Health of Adolf Hitler


• 
  Hitler's Last Will and Testament - English


• 
  Hermann Göring


• 
  Heinrich Himmler


• 
  Joseph Goebbels


• 
  Grand Admiral Karl Dönitz


• 
  Paul Von Hindenburg


• 
  President of Germany - Reichpräsident


• 
  Chancellor of Germany - Reichskanzler


• 
  Dr. Theodor Morell


• 
  Dr. Felix Kersten


• 
  Joseph Kessel


• 
  Tabes dorsalis


• 
  Operation Valkyrie


• 
  General Friedrich Olbricht


• Major General Henning von Tresckow


• Colonel Claus von Stauffenberg