Let $f(z)=\frac{1}{\sqrt{1+z^4}}$. The branch points are $e^{\frac{2k-1}{4}\pi i}$. I am going to find ALL possible branch cuts.
When $z$ traces a closed curve (anticlockwise) around any of the above branch points, the complex number $1+z^4$ also traces a closed curve around the origin. Therefore, the argument of $f$ increases by $\pi$. So we can cut $\mathbb C$ by four curves connecting each branch point to $\infty$.
Question: Are there any other ways to arrange those branch cuts? For example, I find the cuts by the intervals $[-e^{\pi i/4},e^{\pi i/4}]$ and $\{z\in\mathbb C:\arg z=\frac{3}{4} \pi\}$ also possible. How can I find ALL possible ways to do the branch cut?
I hope that the answer can be more formal and rigorous than what I do above.