I think that's a reasonably plausible scenario given that the technologies and skills necessary to 'turn someone immortal' will probably* not come cheap - at least at first.
I don't know whether a society like that would survive for long. The sight of a wealthy elite granting themselves immortality while the rest of us age and die would surely be too viscerally unpleasant to pass without generating a huge build-up of resentment, even among those people who are otherwise not normally that upset by wealth inequalities.
I've often wondered what first contact with extraterrestrial life would look like and what sort of consequences it would have for human society, but it's much harder to say with any degree of confidence what will probably happen if/when we 'invent immortality'. There are too many variables to weigh up.
As it stands, I'm of the pessimistic slant on this one, but perhaps by the time we get there we'll have long since solved poverty, overpopulation, climate change, resource depletion, famine, water pollution, etc. And by that point, maybe immortality would be a more straightforward affair.
But I can't see us being ready for such a leap for many decades yet, and if I can be a total luddite about this, I actually kind of hope it doesn't happen during my lifetime. It's an endless can of worms.
* Then again, who really knows? This research is at an incredibly early stage relative to the prospect of preventing human aging.
You all know what this means, right? Now we're going to have vampire mice running around sucking each others' blood to stay forever young. And the cats will eat the mice and start to figure it out, and then we'll have vampire cats. And any cat owner will tell you how little value cats place on human life, so guess who they're gonna feed on?