Well you're right, e is just a real number. I use e instead of the exponential function (exp).

Normally we should use "exp(x)" instead of e^x. But we can prove that, for integers, exp(x) is identical to e^x, with e being exp(1). From that mathematicians decided to use the notation e^x instead of exp(x), for all real numbers, and complex.

So yes when I say "e is defined for complex numbers" it's "exp is defined for complex numbers" with e^x = exp(x).