The term "imaginary" for referring to the so-called "imaginary numbers" (which, by the way, are only a one-dimensional sliver of the full complex numbers that are actually used here) is a historical artifact that has caused way more confusion than it should, now. Let me say one thing that is crucially important here:
There is no ontological difference between "real" and "imaginary" (or better, "complex") numbers.
No matter what type of mathematical philosophy you accept as to what the ultimate ontological status (i.e. whether or not and in what way mathematical objects are "real") of mathematical objects is, there isn't one major such one I've found that somehow pins these two, but no others, as having distinct ontological statuses.
Mathematics in its fullest deals with a huge variety of objects, many of which are far stranger and more abstruse than "imaginary numbers", but they don't seem to provoke the same kinds of reactions, although maybe that's just a function of that most who get to the point of being able to understand them will have to have long gotten over this notion. For example, right here with the context of quantum theory: forget complex numbers, you have an infinite-dimensional quantum state vector - how "real" is that kind of thing? Why does it not raise even more eyebrows in the same sense regarding its mathematical ontology (as opposed to its "physical" ontology)? Yet it's useful for creating a language and narrative we can use to describe our world in a very precise sense. The same goes with the complex numbers, and the same also even goes with the "real" numbers.
The whole "confusion" about this, as much as I can see, results entirely from the name, and the reason for that name was because in earlier times, before modern mathematics, there was a tendency to look at new mathematical objects with suspicion, and this goes back even further - the negative numbers, for example, were once called "absurd numbers" (in fact, negative numbers and complex numbers were actually both subject to such challenges at the time the latter was invented, though the former had been far earlier in ancient China), and if we go back even further to ancient Greece, the irrational numbers were viewed similarly. And let's not forget zero, and even one.
Mathematics in these eras was much less rigorous, very conservative, and had different standards (to the extent you can talk of "standards" at all) regarding what was/wasn't "acceptable" as compared to today's mathematics.
For modern physics, a system of complex quantum state vectors is the most elegant formalism we have so far for the role it plays in quantum theory. Complex numbers can be defined simply as pairs of "real" numbers:
with suitable operations (and perhaps suitable "data type"), and hence are they really all that much less "acceptable", here, than real numbers, matrices, and the idea of an infinite dimensional vector (which can be thought of as having infinitely many components) with other operations defined on it?