First, the idea of biological immortality doesn't really work. You can't fight entropy. All things degrade eventually. The most typical work arounds I've seen in SciFi involve slowing the aging process combined with organ transplantation. Though there are some species who can rejuvenate, and people speculate on applying this to humans. The big impediment to this is the brain. It's not an organ that you can replace, and who can say what effects rejuvenation would have on memory or personality?
You could throw in "The Singularity", or the point at which computers can completely simulate a human brain. Then you upload your consciousness to the cloud. From there you can stay digital or download into a fresh brain. However, even if you could non-destructively capture the entire information in brain, then you're left with two consciousnesses that both think are the genuine article. The original still dies at some point.
Let's get past that and say that you can really be biologically immortal without any intermediate steps involving your destruction or the death of your original body. Who can say what the psychological effect of grand amounts of time would have on a human mind? Heinlein tended to think it led to despair and apathy which can only be broken by increasingly more novel and taboo experiences. Not to mention the society pressures. Immortal people still take up space and resources, so they'd still need to work. Easy to see how it would be a beating to work full time for eternity, or how society would put on expiration date on people eventually to cycle the population artificially.
Personally, I just wouldn't opt for the immortality treatment in the first place. This world is great and everything, but I wouldn't want to spend forever in it. Especially not when true happiness is waiting on the other side
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