Discussion on whether the notion of complex numbers do exist

It is known that complex numbers, complex geometry, complex algebra etc. have become common mathematical tools for many scientists, physicists, and mathematicians for long time. Nonetheless, among serious mathematicians, still there are people who question if such entities do exist in nature, or are they only in the mind? For these authors, we admit that complex numbers are convenient tools to work out physics problems, but we shall be cautious to come up with complicated new mathematical structures, even if we hear such “generalization” method can be useful for finding the new physics (cf. Dirac). On the contrary, it is often the case that going simpler and simpler will lead you to new physics; for instance, instead of come up with monster group etc., perhaps one can begin to consider low dimensional geometry (cf. William Thurston, Three-dimensional geometry and topology, Princeton Univ.
Press, NJ, 1997). Perhaps, the real space of crystalline corresponding to complex 3D geometry can be found useful in the near future.