Something I've noticed is even older chess puzzles never seem to age, somehow they even seem more masterful and demanding then the puzzles created today, but what if the solution for a puzzle is refuted by a program such as AlphaZero, or when quantum super computers are made, stronger engines, different engines?Will there be any shift with established puzzles, or do we just get to enjoy new theory and different puzzles?
Our modern chess engines are already perfect for solving tactical puzzles. @Bad_Bishop's example is an easy problem for Stockfish.
If you're talking about artificial deep chess problems (not puzzles), even AlphaZero might have problems. Some chess engines (e.g. Houdini) has a "tactic" mode for those problems.
No. Nothing much will change with stronger engines.
Most probably a wide majority of cooked puzzles from history are already within the grasp of today's computers (and even of computers from, say, 2005), and that's why new softwares shouldn't bring any major changes, but it doesn't mean that they wouldn't bring some marginal corrections.
A funny example is given by Tim Krabbé in entry 324, "A cooked, correct study", of his chess diary.
[FEN "8/8/7r/3N4/8/8/7P/R3K2k w - - 0 1"]
White to play and win - E. Pogosyants, EG 1978
From the human point of view, the single solution is the composer's intention: 1.Ne3 Rxh2 2.0-0-0#.
From the tablebase point of view, since castling is not implemented (at least, such was the case in 2006, it might have changed...), this solution is impossible. However, there is also a single winning move: 1.h4! after which the KRNKR endgame is a mate in 33. Less poetic, sure.
So this is a study with two solutions. A human solution that is beyond the grasp of the tablebase, and a tablebase solution that is beyond the grasp of humans.
The main but pointless point is that the puzzle has changed because of the computer assessment.