[@fdeviant]:  The easy answer is both.

The hard answer is that Odysseus is an alias because he is not the Legendary King of Ithaka from the epic poems of Homer.  However, at the same time, his family upholds a tradition of being descended from the heroes of Greek Mythology, which allows him to be a slightly enhanced "Badass Normal".

I actually have a character sheet for the Immortals in the works.  I just have not gotten around to finishing it.