I have seen a lot of arguments recently where, for instance: $$ \sqrt{z^2-1} = \sqrt{z-1} \cdot \sqrt{z+1} \hspace{5mm} (*) $$ without ever specifying the chosen branch or the chosen branch cut (these expressions were later used in integrals so I am fairly sure that we need to make them single-valued and we are thus not referring to a simple equality between sets of values).
I would like to know if I am overlooking something fundamental when I am saying that $(*)$ is not correct (or at least not complete without specifying the branch and the branch cut) if we want to integrate say $\sqrt{z^2-1}$ itself.
Moreover, I would like to know how can I evaluate $\sqrt{z^2-1}$, having chosen the branch cut $z \notin [-1,+1]$ and the positive real part branch, as $z$ approaches the cut from below and from above (I can easily do this if I only had $w = \sqrt{z}$ but now it seems to me that there is "another plane" to consider between the z-plane and the w-plane before I can evaluate the expression).