In answer to
"Why do we need to study numbers which do not belong to the real world?"
you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe.
I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check. (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes. After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.)