Reading a paper by Manfred Clauss called "Deus praesens: Der römische Kaiser als Gott" (Deus praesens: the Roman Emperor as God), in which he critizises modern scholars' tendency to downplay or outright deny that the Romans worshiped the Emperor/Cesar as a living god, i.e. during his lifetime.
Romans did indeed worship living men as gods in many cases even before Cesar, so there is a certain tradition there. But it all went into overdrive with Cesar. Clauss shows that this took place not only in the eastern Empire, which is often said, although it was perhaps more pronounced there (and it's there that the Epiphany holiday apparently took shape): in Rome, too, Cesar was worshipped as a god, and even became a god officially, sanctioned by the state. That is, he was elevated among the "State Gods" even while he was living.
Even in 49, after Cesar crossed the Rubicon, he was greeted as a god in the municipia. After his victory over Pompeius (48) and even more after his vicotry near Zela (47), he was worshiped as a god in the east.
There were of course skeptics in Rome and later detractors, but this is to be expected. Cesar was probably worshipped more in the east than in the corrupt capital, but it speaks volumes that even there the anti-Cesar forces had to accept it. Think Trump vs. Washington DC, except that Cesar actually did clean the Swamp, and was reverred for it (and other reasons).
The paper has lots of interesting things to say, including about the time after Cesar, maybe I'll translate a few passages.