People are afraid of death ... not that this makes any logical sense, mind, but still, they are. I expect it's evolutionary: in order to breed and continue the species you have to make an effort to not die before that can happen, so we have an aversion to death in general programmed into us because of that.
I am not afraid of death, but I am afraid of dying, to a degree. I really don't want to go through all the pain and problems that come with getting old and becoming dead. I don't want to lose my sight, hearing, mobility, or memory for no reason other than I didn't die early enough. From that perspective, you could say that I welcome an early death, but not too early because I haven't experienced as much of the world as I would like to. Not that I'd care when I'm dead, but still.
I'd rather not be immortal. However, I think it would be cool to stop aging after a certain point. Like, once you turn 25, your body just continues to be that age for the next fifty or sixty years, and then you die of natural causes. Going off from that, I think it would be absolutely terrible to continue aging, yet also be immortal. Your body would continue to break down forever, and sooner or later you would be left a vegetable, still technically alive but void of any ability to do anything. That would be just horrifying.