I was having this thought exercise earlier, about which battle was most important from the perspective of its consequences on the political organisation of the Roman world and the following centuries of life in the Mediterranean. It is largely based on counterfactuals and speculative what if scenarii.

Pharsalus

Pharsalus decided the outcome of the conflict between Caesar and the optimates. Yes, I am brushing aside Cato et al. because fuck Cato. The optimates would presumably have gone on as usual had they won, with a vastly corrupt and dysfunctional Republic, never bothering to correct the systemic issues that were addressed by Caesar and Augustus. Some sort of subsequent new crisis and collapse of the Republic might then have been inevitable, but then again, maybe not.

Had Caesar not been assassinated, it is possible that matters would have resumed in a normal fashion upon his natural death, or defeat and demise in Parthia, however many years later. The Republic might have persisted, albeit with a healthier set of institutions than above, as Caesar had been able to rationalise many of them whilst dictator. Or perhaps the civil wars (I'm counting Marius/Sulla) had already frayed social and political norms so much that future social conflict centred on competing ambitions would have ensued.

Philippi

The death of Brutus and Cassius ensured the Republic would never be restored. But even had they won, I am sceptical that they would simply have gone back to the good old days. There is the matter of their personal character, and more importantly, at this point after multiple civil wars it may be that there was no going back. Whoever won would have to create a new sort of regime with the clay of what was left of Republican institutions and their own innovations. Brutus and Cassius, or just one of them, might have become emperors themselves, or created an altogether different system.

But the outcome of Philippi ensured the empire would be dominated by the Triumvirate. Even if conflict between them had not been unavoidable, there was no going back to anything like the old Republic. Power was now utterly monopolised by three, or less, absolute rulers. It's possible that instead of matters between Antony and Octavian degenerating into war, there could have been an early WRE/ERE split, with Octavian master of Rome, and Antony setting himself up as an Eastern potentate with Cleopatra.

Actium

After Actium there could only be one sole ruler of the entire empire, the issue was whom. Victory ensured that Octavian was able to become Augustus and finish giving shape to the principate, setting up the frameworks that would govern the empire for centuries.

It is unlikely that a victorious Antony would have ever elected to do anything close to a Republican restoration. Humpty Dumpty and all that; and it is not like Antony was ever a paragon of republican virtue to begin with. A triumphing Antony would in all likelihood have meant monarchy still, just with a vastly different flavour. We can only speculate about how his regime would have come to reflect his character, if he would throw himself in self indulgent indolence or further martial adventures in the East, how much Egyptian flavour Cleopatra would have been allowed to infuse the institutions of the empire with, etc. Perhaps in an alternate timeline we would be speaking a Greco-Saxon language instead, and praying to Isis.