With real numbers, we have the Power of Power Law:$$a^{xy}=(a^x)^y.$$
However, this doesn't always work with complex numbers. For example, Mathworld states that: $$(i-1)^{2i}\neq[(i-1)^2]^i.$$
I was wondering if there might be an intuitive explanation for why the Power of Power Law (sometimes) breaks down in the complex case.
Consider how we may prove the Power of Power law for positive reals. First we must define the exponential function $$\exp x = e^x = 1+x+\frac12x^2+\frac16x^3+\cdots$$ And let $\ln x$ be its inverse (this exists because $\exp x$ is bijective). Suppose that $$e^{ab}=(e^a)^b.$$ Then $$a^{xy} = (e^{\ln a})^{xy} = e^{xy\ln a} = (e^{x\ln a})^y = (a^x)^y$$ as desired. Now, it remains to be shown that $e^{ab}=(e^a)^b$. Call the LHS $f(a,b)$ and the RHS $g(a,b)$ and treat $a$, $b$ like variables. It is easy to verify that their partial derivatives with respect to $a$ are equal, $f_a=g_a$, and similarly for that with respect to $b$, $f_b=g_b$. So these functions at most differ up to a constant, which clearly must be $0$, as $f(0,0)=g(0,0)$. Thus, the conclusion is proven.
This proof, however, breaks down when $a\in\mathbb C$. This is because the complex logarithm is infinitely multivalued, so we cannot do the manipulation we have done above.