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?
The complex numbers are just a distraction. We can rewrite your example as

$$(-1)^2=1 \implies \big ((-1)^2 \big)^{1/2}=1^{1/2} \implies-1 = \sqrt 1 = 1 \implies -1 =1.$$

Do you see where we made the mistake above?