For the most part, eternal life isn't all it's cracked up to be. Indeed, you will remain ageless while all those around you - your friends, family, and everything you love - gets old and eventually dies. But as with any rule, there are exceptions.

There are no doubt some people who would happily accept this burden, making the most of it. These would typically be people who don't have any loved ones to lose (or exceptionally few) nor any particular attachment to anyone or anything. Even I'm thinking such people wouldn't exist, but there [i]are[/i] 7-billion of us and our population, unfortunately, is constantly increasing. Somewhere, such a person would exist. It would come to the question of what they would do with their eternal life. Love is clearly out of the question (and like that, I bow humbly from eternal life) as are any strong forms of emotional attachment. As the years passed, this immortal would retain the knowledge of his/her age and ones yet to come. They could do anything with their lifetimes of knowledge. And I mean [i]anything[/i]. Conquering warmonger who becomes an everlasting dictator of the world? Why not? Perhaps they could become a wise guru-type person instead, passing their wisdom down from their time to the future. Or maybe they'll just run a bar for all time or something like that ^_^;

In summary, the concept of eternal life is currently that and only that. A concept. As such, you would probably get different answers from each person you asked, since despite our staggering world population, no two humans think [i]exactly[/i] the same. I myself would not take eternal life, but that doesn't mean I don't want to live forever. I'd like very much to live forever, just not as I am ^_^ but more to your question. Were it me given the choice to remain with a mortal or leave them, I would stay. Just because a person dies doesn't mean they're gone, if that makes sense. In fact, my grandmother's birthday was just yesterday. She may not have been able to properly celebrate it, currently being a pile of ashes in a black box above the kitchen sink, but a person isn't truly dead until there are no more people in this world who would care to remember them. To quote one of my favorite songs to this day:
[quote=Megadeth]Smile when you think of me. My body's gone, that's all.[/quote]