Comment by ccppurcell

C is the only way to make a field out of pairs of reals. Also (or rather just another facet of the same phenomenon) we might be interested in polynomials with integer coefficients, but some of those will have non integral roots. And we might be interested in polynomials with rational coeffs but some will not have rational roots. Same with the reals but the buck stops with the complex numbers. They are definitely not accidental they are the natural (so to speak) completion of our number system. That they exist physically in some sense is "unreasonable effectiveness" territory.