Why can't I combine complex powers
Even without any complex numbers: $-1=(-1)^{2\cdot\frac12}\ne((-1)^2)^{\frac12}=1^{\frac12}=1$.
But you're right, the problem is that raising to a (non-integer) power is essentially a multivalued function.
This is related to a note by Euler, maybe he was the first to realize that $i^i$ is real. Actually, $$i^i = (e^{i\pi/2})^i = e^{-\pi/2}$$ on the other hand $$i^i = (e^{i(\pi/2+2\pi n})^i = e^{-\pi/2 -2\pi n},\ n\in\mathbb{Z}.$$ So maybe it is better to say that $i^i$ is a subset of the $\mathbb{R}$ and that certain equality signs are to be understood as congruences.