Was Julius Caesar ambitious?

In Shakespeare's classic play, Julius Caesar is portrayed as a shrewd politician who is clearly egotistical and domineering. Despite being a self-centered narcissist, Caesar's ambition remains ambiguous, which emphasizes Brutus's internal struggle to join the conspirators or reject their proposal. Brutus's decision to join the conspirators relies solely on his belief that Julius Caesar is ambitious. In act two, scene one, Brutus voices his concerns regarding Caesar's ambition in a moving soliloquy, where he likens Caesar to a "serpent's egg," which will become dangerous after it hatches. Despite Brutus's belief that Caesar is ambitious and motivated to become emperor, there is no clear evidence that unquestionably proves that Caesar is ambitious.
In act one, scene two, Casca informs Brutus and Cassius that Caesar reluctantly pushed the crown away when Mark Antony offered it to him in public. Although Casca tells them that Caesar "was very loath to lay his fingers off it," one cannot trust his opinion as solid proof that Caesar desired to be crowned king.
However, one cannot deny the fact that Caesar is an egotistical, self-centered man. He constantly refers to himself in the third-person, says that he is more dangerous than danger itself, and compares himself to the North Star. He also dismisses the soothsayer's warnings, rejects his wife's advice not to go to the Capitol, and exercises his authority by refusing to restore Publius Cimber's citizenship.
Despite his confidence, narcissistic personality, and domineering nature, one cannot definitively prove that Julius Caesar was ambitious, which is what makes his assassination and Brutus's participation dramatic and intriguing.

Shakespeare downplays Caesar's ambition right up until the moment he is attacked by the conspirators. In the first two acts of the play Caesar is offstage much more than he is on. When onstage, he is portrayed as a sort of portly politician who is acting democratic, friendly, cordial, and humble in order to make as many friends as possible among the patricians and to make a favorable impression on the plebeians. Shakespeare evidently intended to have Caesar show his true self just before he is assassinated, which would contribute to the dramatic impact of that bloody scene and would prove to the audience that the conspirators might have been justified in plotting his death. The fact that Caesar knows he is going to be made king has gone to his head. He disregards his wife Calpurnia's pleas to stay at home, the warnings of the Soothsayer to "Beware the Ides of March," and all the supernatural omens that seem to have been sent as warnings. He also insults all the petitioners surrounding him, getting them to grovel and then expressing his contempt for their groveling. Perhaps he feels that his coronation is all but certain and now he can stop being so cordial and democratic. Just before he is stabbed to death, he makes his most revealing speech:
I could be well moved, if I were as you;
If I could pray to move, prayers would move me;
But I am constant as the northern star,
Of whose true-fix'd and resting quality
There is no fellow in the firmament.
The skies are painted with unnumber'd sparks;
They are all fire and every one doth shine;
But there's but one in all doth hold his place.
So in the world, 'tis furnish'd well with men,
And men are flesh and blood, and apprehensive;
Yet in the number I do know but one
That unassailable holds on his rank,
Unshaked of motion; and that I am he,
Let me a little show it, even in this;
That I was constant Cimber should be banish'd,
And constant do remain to keep him so.
And when Cinna starts to remonstrate, Caesar says,
Hence! Wilt thou life up Olympus?
Then they all close on him with swords and daggers. Though we know it was planned beforehand, it still almost seems as if Caesar provoked them into doing it.
When Caesar says "If I could pray to move," he seems to be implying that he does not believe in prayer, possibly that he is above prayer because of his distinction. He cannot pray to the gods because he secretly feels like he is a god. This conviction is further implied when he says, "Hence! Wilt thou lift up Olympus?" The summit of Mount Olympus was the home of the gods, and Caesar evidently imagined himself up there among them, drinking nectar and eating ambrosia. Caesar is so ambitious that he won't be satisfied with being a mere king. The next step up what Brutus called "young ambition's ladder" is to become a god. That would not have been hard to achieve in ancient Roman times. The Senate could declare a person to be a god and order everyone in the empire to worship him or her as such. Caesar's successor Octavius Augustus became a god by Senate decree, as did Caligula. Augustus Caesar's fiendish wife Livia was declared a goddess posthumously on the orders of Emperor Claudius. Julius Caesar himself was also made a god posthumously by public acclamation.
So perhaps even Cassius and Brutus did not fully comprehend how ambitious Julius Caesar actually was. When Antony makes his famous funeral address, he does not seem to deny Caesar's ambition because it was all too obvious. Antony may not have realized that his good friend's ambition went beyond being a mere mortal.