Personally I'm going for the non-highlander type of Immortal, The movies and such were kinda meh in my opinion. I'm going for the more "superman" (though no ridiculous strength or flight or lazer eyes) type of Immortal who literally cannot be harmed, watching the world go by, being in but not really a part of it. That's what I'm shooting for at the beginning of the book.
My character is like that and part of a whole world of weirdness involving a time before the universe existed and so on.