The fact that there are three cube roots is precisely the problem.
If std::cbrt(z) were defined for complex z, one would expect it to yield std::exp(std::log(z)/3) (note that std::log returns the principal value). The problem with this is that if z happens to be a negative real, this will not give the same root as the std::cbrt function for reals (namely the negative real one). It will give the one that is inclined at an angle of +pi/3 from the positive x-axis.
That means there's no good way to define std::cbrt(z) for complex z: it violates either one expectation or the other.
I suspect there is no mainstream programming language that has two overloads of the cube root function---one with domain and codomain R, and one with domain and codomain C. They had to pick one---probably the real one, because the complex one you can emulate using whatever is the equivalent of the complex `pow` function. In the case of C++, the choice has been made for us already in any case.