> Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.

Not true historically -- the origin goes back to Cardano solving cubic equations.

But that point aside, it seems like you are trying to find something like "the true meaning of complex numbers," basing your judgement on some mix of practical application and what seems most intuitive to you. I think that's fruitless. The essence lies precisely in the equivalence of the various conceptions by means of proof. "i" as a way "to do arbitrary rotations and scaling through multiplication", or as a way give the solution space of polynomials closure, or as the equivalence of Taylor series, etc -- these are all structurally the same mathematical "i".

So "i" is all of these things, and all of these things are useful depending on what you're doing. Again, by what principle do you give priority to some uses over others?

Maybe.

You disagreed with the parent comment that said

"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."

I see Complex numbers in the light of doing addition and multiplication on pairs. If one does that, rotation naturally falls out of that. So I would agree with the parent comment especially if we follow the historical development. The structure is identical to that of scaled rotation matrices parameterized by two real numbers, although historically they were discovered through a different route.

I think all of us agree with the properties of complex numbers, it's just that we may be splitting hairs differently.