Are Imaginary Numbers Really “Imaginary?” I find the naming convention of “Imaginary” misleading, as it does give a sense that the quantity is merely an abstract construct used to mitigate the difficulties of performing some mathematical operations.
My question is, other than the wavefunction for example, where else do complex numbers have physical significance?
Also, since only $|\Psi|^2$ has physical meaning i.e. probability amplitude, does this really mean that the “Imaginary” part is really significant, or just another purely mathematical construct?
 A: I believe that you can do all of physics apart from QM without using complex numbers: complex numbers are a convenience (generally because $e^{ix} = \cos x + i \sin x$), but they are only a convenience.
However if you want to do QM you either end up using complex numbers or creating mathematical objects which have all the properties of complex numbers: you can't do QM without such mathematical objects.  In particular while you measure $|\Psi|^2$, the evolution of the system depends on $\Psi$: you can't do QM without dealing with complex quantities.
So, I believe the answer is that they only occur necessarily in QM.  However since QM is our most fundamental theory of physics that's not a small statement.
I would be very happy to be wrong about either point.
A: 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:
$$(a, b)$$
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?
A: One could say that all numbers are imaginary in that sense, they are abstract concepts that we force on a relation into nature just to allow us to have some descriptive power of our surroundings. The fact that natural numbers seem more "natural" or easy to relate to has nothing to do with what they really are; abstract ideas without physical meaning until we, humans, impose a one-to-one relation between them and material objects.
On the other hand if you consider real numbers, naturals, rationals and irrationals to be "real" entities (whatever that means) then you must be sincere and include imaginary numbers on that grouping because all numbers are as artificial as they are, a product of logical abstract argumentation and a consequence of a set of axioms that have no more fundamental justification to be there.
So, your question is more a philosophical one. What is real? Are ideas real for you or is the material world the only cause protagonist of reality? This is a complex topic with many many historical opinions that can be superficially encapsulated between two main frameworks; idealism versus materialism.
Physics as a branch of natural philosophy is materialistic in its essence while mathematics have come in idelistic (even platonistic) flavours throughout the years. So when asking if any imaginary number represent an observable quantity the answer is difficult and depends on your philosophical stand and also on the relation between physics and mathematics (which is not so trivial and was rejected by some philosophers like Giordano Bruno before Galileo established it as the main descriptive language of nature).
Are negative numbers "real"? That's the same kind of question. You can observe negative temperatures right? Well, but that is a convention that helps to understant this variable in relation to a reference. Are there any real life/world examples of negative numbers?
The same goes for imaginary numbers. You need imaginary numbers to describe the fundations of Quantum Mechanics, you need imaginary numbers to understand many situations in Electronics, you need them even for any Fourier Analysis so you need them for the study of sound waves, electromagnetic waves and even mechanical oscillations in many problems. Our modern world is impossible without the discovery/invention (depending if you are idealistic or materialistic) of imaginary numbers.
There are many problems where complex numbers help to reach an answer quickly but that have ways of solving them that may not rely on complex numbers (for example, you can describe a wave with sines and cosines or with complex exponentials and for very simple situations they are both possible pathways to analytic answers), but there are also problems with solutions that are unreachable if complex numbers are not used. That is interesting; if they are really needed to solve some problems and not only usefull to solve them in a simple way, are they "real"? Is their relationship, as ideas, with the material world a real thing?

Also, since only $|\Psi|^2$ has physical meaning

By the way, here you still need complex numbers, even if the probability amplitude is a real number, if you ignored the complex solutions the probability amplitude would be another completely different real number and QM would not explain anything about the real world.
A: It is possible to do all computations that traditionally involve complex numbers, without reference to imaginary numbers at all.  An equation written in complex notation can always be separated into two equations in real notation.  Example:
$$Z_1^2 +Z_2^2 = C$$
is the same as $$x_1^2 -y_1^2 + 2i(x_1y_1) + x_2^2-y_2^2 + 2ix_2y_2= C_r +iC_i$$ which is exactly the same as the pair of equations $$x_1^2 -y_1^2 +x_2^2 -y_2^2= C_r$$ and  $$2x_1y_1+2x_2y_2 = C_i$$
However, complex notation using i, and especially i in an exponent,  greatly simplifies the notation and provides a useful way to visualize the meaning of the computation.
