Say we got a multivalued function $$\log(z^2-1)\equiv\log(\xi(z)).$$ Usually, we would choose principal values such as $\arg\xi(z)\in[0,2\pi)$ or $\arg\xi(z)\in[-\pi,\pi)$. In order to find the branch cuts, for instance, for the $1^{\text{st}}$ one, we'd then need to impose $\arg\xi(z)=2\pi k$, namely, $$\mathfrak{Re}\{\xi(z)\}=x^2-y^2-1>0$$ and $$\mathfrak{Im}\{\xi(z)\}=2xy=0.$$
From both these conditions we'd finally find that the branch cuts are described by $y=0$ and $|x|>1$ (basically, $\mathbb R-(-1,1)$).
I then thought whether I could use the same methodology to find branch cuts for other, less common, principal values such as $\arg\xi(z)\in\left[-\frac{7\pi}{4},\frac{\pi}{4}\right)$:
We'd impose $\arg\xi(z)=-\dfrac{7\pi}{4}+2\pi k$, that is, $$\mathfrak{Re}\{\xi(z)\}=x^2-y^2-1>0$$ and $$\mathfrak{Im}\{\xi(z)\}=2xy>0.$$
We can finally conclude that $\sqrt{x^2-1}>|y|$ and that either $x,y>0$ or either that $x,y<0$, meaning, if I understand correctly, that the branch cut isn't just a line, but a domain? How'd something like this be represented graphically? Would this be it? https://www.desmos.com/calculator/rr8nkw3a8c . Does all of this make any sense at all?