Exponentiation of complex numbers by complex numbers is tricky, and ultimately solved "the right way" by allowing Riemann surfaces other than $\mathbf{C}$ to be domains of the intervening functions.
If I were to exponentiate a complex number $$z = \rho\cdot e^{i\theta}$$ by a real number $t\in\mathbf{R}$, I would just set:
$$z^t := \rho^t\cdot e^{it\theta}.$$
Am I missing something or is it that simple? Is the function $z\mapsto z^t$ continuous/holomorphic? and on which domain of $\mathbf{C}$?
It sure recovers the usual exponentiation by $t$ if $z$ is real.
Thanks.