"Why do we need to study numbers which do not belong to the real world?"
Having been through the relevant mathematical mill, I subsequently engaged with Geometric Algebra (a Clifford Algebra interpreted strictly geometrically).
Once I understood that the square of a unit bivector is -1 and then how rotors worked, all my (conceptual) difficulties evaporated.
I have never had a reason to use (pure) complex numbers since and I suspect that most engineering/physics/computing types would avoid them if they were able.
Likely you have the above group mixed together with pure mathematicians that feel quite at home with the non-physical aspects of complex numbers and wouldn't dream of asking such an impertinent question:-)
