Question: I've just stumbled across $$\left(e^{2\pi i}\right)^{\frac{1}{2}}=\sqrt 1=1\neq -1=e^{\pi i},$$do I have some error in my thoughts there or does $(a^b)^c=a^{(bc)}$ not hold for complex numbers?

Answer: 
Actually, that equality *makes no sense* for complex numbers. How do you even *define* $a^b$ if $a$ and $b$ are arbitrary complex numbers?
That's because
$$
(e^{2\pi i})^{1/2}\neq e^{\pi i},
$$
even though their square is equal:
$$
(e^{2\pi i})= (e^{\pi i})^2.
$$
Just like what's in real numbers, every complex number have two square roots. The power $1/2$ placed at the top right corner just gives you one of them. $e^{\pi i}$ is another square root of $1$.