I feel I have got more out my readings this time. It is an interestingly rhetorical play (hence, possibly, the many quotes ones knows from it). The set pieces are magnificent and it is more a play of declamation than dialoguic interaction. This was the Rome of Shakespeare's imagination and is not unlike that of Coriolanus.
Thinking of it as either the tragedy of Brutus or a History of Rome gives one a better feel for the play too. Caesar is set up as a potentially tragic figure (with common enough human traits) only for his death not to be tragic. He dies because of the beliefs, thoughts, envy, etc of fellow Romans and Brutus, as a tragic figure, is indeed deeply flawed. Vacillating, not quite stable (the two versions of the news of Portia's death show Shakespeare regarding him as a man who could go either way).
Superstition also plays an interesting role I hadn't noticed before. Lauded, castigated, condemned, finally cited as the cause of Brutus' suicide. There is no sign of this in Antony and Octavian and though they have their differences it is clear Octavian is the dominating character whereas the lack of cohesion between Cassius and Brutus contributes to their failure.