I've always found it slightly amusing that we've now reached the time in the history of civilization where immortality, even were it possible, has become a massive liability. One can no longer simply assume a new identity of proper age, and attempting to do so is liable to get you jailed, or worse still, in some shady government lab.
It's even getting to be the case now that you can't just move to a new country and start over. Unless you've got skilled forgers close at hand, you're going to have a bad time immediately, and with facial recognition and biometrics, you'll have a worse time still.
It's perhaps the definition of a low-probability corner case, but if any alchemists happened to succeed, they're no doubt having a pretty depressing century.