Are we answering the right question?
You touch upon an interesting point, but I have the feeling that your question isn't specific enough yet to reach a proper resolutions. Others have argued that 'complex numbers' aren't necessary for quantum mechanics. While I agree with their arguments, I think they're answering the question
Do we need something we call 'the complex numbers' to describe Quantum Mechanics (QM)?
and answer that, no, we can use some other mathematical object that isn't called that instead.
But that is a complicated answer to a trivial question, as I can simply define the 'lizard numbers' with exactly the same definition as the 'complex numbers' (without using that name, of course) and say you can simply describe QM using 'lizard numbers' instead. You might say that I'm cheating, but am I also cheating if my lizard numbers are different from complex numbers, but not very and can still be interchanged with the complex numbers to yield a valid theory of QM?
For example, suppose my lizard numbers extend the complex numbers with an
l in addition to the
i that indicates the 'lizardly axis' (in addition to the real and complex axis) but is usually set to
0 when performing QM, as there are no lizards on when working at a quantum scale (The lizardly axis is integral, as fractional lizards are animal cruelty).
Clearly, there are some issues that could be captured by asking better questions. An approach is this:
Is it possible to describe QM without using a mathematical structure that is 'essentially the same' as the complex numbers?
This question appears to represent the problem a bit better. However, it crucially depends on 1) what 'essentially the same' means and 2) what is a description of QM, or what is a physical description in general.
When are two mathematical objects 'essentially the same' for QM?
I think that you would agree that my lizard numbers yield an description of QM that is 'essentially the same', as I can simply replace every complex number by a lizard number and can keep the rest of the description. In the context of QM, it's not much more than a renaming, really.
But can we give a precise definition? If we are working within mathematics, I might come up with an approach. But we aren't in the realm of mathematics, but in physics and physics has some (mathematical!) problems that are 'widely regarded to be true' for which there is no mathematical proof (yet?). Take for instance the Yang-Mills gap hypothesis. The hypothesis has been confirmed to be sound by physical experiments and is part of standard theory, but this doesn't satisfy a mathematician (and perhaps some physicists), as this doesn't lead to a mathematical proof.
As we have seen that something can be proven in physics without proving it in mathematics, we really need a definition in physics. My knowledge on physics is lacking, so I'm unable to proceed here. But I doubt that an expert in physics would be able to give an unambiguous definition of what 'essentially the same' should mean here. (feel free to contradict me on that, though!)
When is something a 'description of QM'?
Contrary to the title, let's look at describing the quantum wave distribution, as that seems easier and is what the question is actually asking.
Still, this is perhaps even harder than the previous point. There exists descriptions of this function in different languages with different terms, so I suppose this should be 'independent of language', somehow. Also, do we take any lecture on this function to be a valid description? Probably not. We likely should require that the description allows us to unambiguously know how interpret the function in the results of physical experiments.
Can we conclude anything?
I hope that I have shown that the assertion that 'complex numbers are necessary to describe the quantum wave distribution function' is not as simple as it seems. Should we ask why something is true, before we know that it is true? Probably not, but then again, I know fairly little about philosophy. Perhaps these tricky questions have easy answers I'm simply ignorant of. If you know them, I'd be very glad to hear them, but this is all I can add.