Not all alchemists are immortal in Harry Potter. However, both Nicolas Flamel and his wife Perenelle Flamel would have been immortal as long as they drank the Elixir of Life which was made from the Philosopher's Stone invented by Nicolas Flamel through his skill in Alchemy. Any alchemist with this stone could theoretically be immortal. However, Nicolas Flamel and his wife Perenelle Flamel died soon after the events of Harry Potter and the Philosopher's Stone as they decided to destroy the stone for the greater good and to prevent it from falling into the wrong hands.