You do in fact need the complex exponential to define this correctly because the function a^x for nonintegers x is only unambiguously defined when a is a positive real number. For example, your function could be either e^(pi i x) or e^(-pi i x), which trace the circle in opposite directions as x varies over the reals. (They happen to agree when x is an integer.)

That's because

   a^b = exp (b ln  a)

That's equivalent to saying, no need for -1 because we have exp.

One can change based of the exponentiation operation. Exp happens to be a convenient base.

If you plot it over which domain?