You can’t properly implement many of the std::complex operations on integers. E.g.,
template <class T>
T abs(const complex<T> &z);
for a complex<long> cannot have T = long return value when complex numbers are represented as (real,imag) pairs, since it returns the value of sqrt(pow(z.real(), 2) + pow(z.imag(), 2)). Only a few of the operations would make sense.
Even worse, the polar named constructor cannot be made reliable without breaking the default constructor and vice versa. The standard would have to specify that “complex integers” are Gaussian integers for them to be of any use, and that one of the constructors is severely broken.
Finally, how would you like your “complex integer division” served, and would you like a “complex remainder” with that? 🙂
Summarizing, I think it would be more sensible to specify a separate gaussian_int<T> type with just a few operations than graft support for integral T onto std::complex.