It wouldn’t really be “us” anyhow. Some thoughts from a political theorist:
The moral philosopher Samuel Scheffler at New York University has suggested that the real problem with a fantasy of immortality is that it doesn’t make sense as a coherent desire. Scheffler points out that human life is intimately structured by the fact that it has a fixed (even if usually unknown) time limit. We all start with a birth, then pass through many stages of life, before definitely ending in death. In turn, Scheffler argues, everything that we value – and thus can coherently desire in an essentially human life – must take as given the fact that we are temporally bounded beings. Sure, we can imagine what it would be like to be immortal, if we find that an amusing way to pass the time. But doing so will obscure a basic truth: that because death is a fixed fact, everything that human beings value makes sense only in light of our time being finite, our choices being limited, and our each getting only so many goes before it’s all over.
Scheffler’s case is thus not simply that immortality would make us miserable (although it probably would). It’s that, if we had it, we would cease to be distinctively human in the way that we currently are. But then, if we were somehow to attain immortality, it wouldn’t get us what we want from it: namely, for it to be some version of our human selves that lives forever. A desire for immortality is thus a paradox: it would frustrate itself were it ever to be achieved. In turn, Scheffler implies, once we’ve reflected carefully on this deep fact about ourselves, we should junk any residual desire to live forever that we might still have. Paul Sagar, “On going on and on and on” at Aeon
Of course, traditional ideas about immortality involve an imperishable world, not simply imperishable people.
See also: Claim: Yes, You Can Upload Your Brain: Fine print: They might have to kill you first
and
What great physicists have said about immateriality and consciousness