I need to know if my intuition is correct here.
Idea: I would use De'Moivre's formula to find all third roots of -2i and exclude one of these rays because these are the values that give $z^3+2i=0$ and $\operatorname{Log}(z)$ is not defined at $0$, hence not analytic at $0$. Given that the third roots of $-2i$ are, $2^{\frac{1}{3}}[\cos(\frac{-\pi}{2}+\frac{2k\pi}{3})+i\sin(\frac{-\pi}{2}+\frac{2k\pi}{3})]$ for $k=0,1,2$. Then pick a cut of principal branch say $k=0$ and say the function is analytic of the set $\mathbb{C}\setminus\{re^{i\theta}:\theta=\frac{\pi}{2}\}$.