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I watched an episode of "The Blacklist" which is a popular show on the TV streaming service Netflix. In the episode, the main character refers to a very tiny marine creature commonly known as the "Immortal Jellyfish". In extreme life threatening situations, an immortal jellyfish is able to revert to a polyp state within seconds. A miracle of nature which is not fully understood. You can read more about it here: https://en.wikipedia.org/wiki/Turritopsis_dohrnii

My question is: if I had a pet named Jimmy and Jimmy is an immortal jellyfish and at some point returns to its polyp state, and eventually transforms back to a mature jellyfish, is it still Jimmy or a different jellyfish?

I'm also interested in how this would apply to humans. If we had that ability and could transform back to an earlier zygote state, would John Doe still be John Doe? There is physical continuity if nothing else.

Welcome to SE Philosophy!

This is what is known in philosophy as a question of identity and is related to the metaphysical discipline of ontology, or the study of what is. In essence, identity is the question of what 'is' is, and is a source of much debate. Are equality and equivalence the same thing?.

Questions of identity related to personhood and the self are addressed in these SE posts.

Questions related to things that change over time are frequently tied to the Ship of Theseus and is old indeed. To wit:

In the metaphysics of identity, the ship of Theseus is a thought experiment that raises the question of whether an object that has had all of its components replaced remains fundamentally the same object. The concept is one of the oldest in Western philosophy, having been discussed by the likes of Heraclitus and Plato by ca. 500-400 BCE.

Note that this isn't really a question of biology, because if a fish becomes a polyp and returns to being a fish, biologists agree it is the same organism. But, are things that change really the same if they become different? With the Ship of Theseus, there are four general responses: yes, no, depends, and it's a meaningless question. That's the universal tell that any answer you get is going to be a result of your metaphysical presuppositions.

This is an interesting question because it is the ship-of-theseus problem in the guise of debate over the nature of biological immortality in the animal kingdom. From an analytically philosophical standpoint, the answer of the question is yes, because it follows from the definition of organisms which is a category which fundamentally accepts the change in state from birth to death. (Biologists would be in trouble linguistically if they had to classify that same living things as different organisms depending on development since there are no clear categories.) In the ship-of-theseus problem, it is less clear because objects are not defined so narrowly in such a way.

Technically, this IS a question about biology. You see, there's a thing called genetics that is very useful in identifying not just species but individuals.

I don't know much about jellyfish reproduction. Maybe they're all "identical twins," for all I know. But is they have genetic variation, then each life stage would presumably have the same genetic code.

Of course, you have to decide whether individual identity is ultimately based on DNA or something else.