Yes, but the reason you're learning those things is that that is the way reality actually works. That's what happens when you actually do experiments. Richard Feynman once remarked: "quantum mechanics was not wished upon us by theorists". We have QM because physicists were dragged to it kicking and screaming by actual experiments that showed them that reality does not work the way classical physics describes. And the reason all those difficult mathematical tools have to be taught is that that's the only way physicists know how to build models that make correct predictions about what happens when you do those experiments.
In other words, yes, you can say QM doesn't make sense, but that's because reality doesn't make sense--at least not if "make sense" means "works the way my human intuitions, which never evolved to deal with quantum phenomena, say it should". Yes, reality doesn't work the way our human intuitions say it should. That means we have to retrain our intuitions--which is what a lot of learning QM is really about.
My teacher told me, that I will really know physics when I will be able to derive my own equations. I extended this definition to "I will really know physics when I will be able to create the physical model of a physical thing, to be able to watch it with my own eyes".
But it's not just complex conjugates, but eigenvectors as well. Also spectacular: "the square root of a negative number, which by definition does not exist"
Uh. That's a complete lack of connection to anything beyond 10th grade math. Maybe not the ideal foundation for criticizing QM.
And yes, I get the complaint that QM comes with a lot of "if you just hold your nose and accept this", and intuition doesn't really work for it - but that's hardly a valid criticism by itself, outside of saying "well, this is hard to understand".
And the idea that a mathematical model isn't perfect and as a result of that can't be true really just betrays a complete misunderstanding how models work.
This piece boils down to "I don't understand it, and so it can't be correct".
Having studied both engineering and physics, I can tell you that there is nothing observable in the universe that requires complex arithmetic/algebra/theory. We use it because it makes the math a lot more convenient.
BTW, the use of complex math for real world applications goes way beyond QM, and predates it significantly. Engineers use it all the time: AC Circuit theory is full of it. As is electromagnetics. As is power (motors, etc). As is anything involving Fourier or Laplace Transforms (i.e. almost all engineering). You can come up with purely real mathematics to describe all of the above, but it would be tedious.
But if their problem is not being able to stomach the "just-because" explanations that every QM resigns to using to explain wtf complex conjugation has to do with reality, then I completely, totally sympathize. Learning QM consists of learning to silence your objections to the fact that none of the things you're being asked to believe make any sense.
It gets somewhat more scrutable when you deal with relativistic QM, where, at least, the terms in Schrodinger equation start to make intuitive sense (it's the first couple terms in the Taylor expansion of E psi = sqrt(p^2 + m^2) psi = m sqrt(p^2/m^2 + 1) psi = (m + 1/2 p^2/m)psi ), but then spinors enter the picture and all intuition goes out the window again.