I view immortality as the ultimate attachment. Death is a natural part of life, not the end but simply a change. Change is good. Immortality is not.
There was an interesting episode in one of the Star Trek shows where a member of the Q Continuum (the realm of immortals with seemingly absolute powers) was put to jail to prevent him from suicide.
A trial, headed by Captain Janeway, was held between him and the Q. At the end, the trial resolved in an agreement that he would become mortal but would be prevented from ending his life immediately.
Indeed, it was the Q (or should I say Q himself) that smuggled poison in to let him end his life which had nothing left but boredom.
As much as we think of our Universe as infinite, I am convinced that, given infinite time within this Universe, one would eventually come to a point of no further growth possible without the change of venue, whether it be reincarnation, going to heaven (which, too, would have to be temporary for it to be meaningful), or any other way.
It is also interesting to note that fictional immortals always have to take other lives to sustain their own: The vampires kill humans and drink their blood, the Highlander immortals chop off each other's head, etc.
Let us also not forget that human existence is riddled with pain, disappointment, and frustration (it certainly has its nice points, too, of course). Who would want that to last forever? Not me!
I find my own mortality to be a plus. As one who believes in reincarnation, I find it much more fascinating to live different lives: As white or black, as male or female, as tall and short, as human or a god or animal, etc.
Variety is the spice of life. Immortality has no spice.