I'm at the end of Academia 500 AF, and haven't beat Jet Bahamut yet (although I know Serah dies at the end).
So my first question is why did he tell Noel to kill him at the beginning of the game if he was immortal and could not die?
And does Caius follow Serah and Noel through the timeline (He first is in A Dying World, then Valhalla, then Oerba, then Yaschas Massif, and ending with Academia 500 AF) or does he just wait, and appear in A Dying World 700 AF after he fights Serah and Noel in Academia 500 AF and the rest of the game has already happened?
He wants to get killed because that way, the goddess (and with her, time itself) will die/cease to exist and so Yule will not be reborn again, just so she can die at a young again and repeat that process.
The Caius in Oerba does say that he is not the Caius that Noel from AF 700 knows so ... maybe with the paradoxes created, Caius also knew that XIII-2 storyline would happen and knew when and where each and every Sera and Noel would appear?
Cloud and Sephiroth - battling each other since '97 for your amusement Give it a rest already!