Is there a list of common identities for principal branches of complex logarithms and roots? Often I have a necessity to simplify some expressions involving roots or logarithms, where arguments appear to be imaginary or even more generally complex. Due to $n$th root and logarithm being generally multivalued, many simple identities, which hold for positive arguments, no longer hold for negative or complex ones.
Thus, for example, when I have an expression like
$$\frac{\sqrt{i\pi+x}}{\sqrt{-i\pi-x}},$$
I have to stop and analyze, whether I really can "extract $-1$" from a radical, and which sign the resulting $i$ will have — taking into account how the principal branch of the multivalued function (square root here) is defined.
So, is there any list of exactly working (i.e. not "up to branch") identities for principal branches of roots and logarithms, which would help simplification of such expressions?

There can't really be any such identities written down in a useful way.
The common ways of writing down things results in expressions that define analytic functions, so if we have an identity $f(z)=g(z)$, the expression $f(z)-g(z)$ will define an analytic function that is identically zero in the domain where the identity is valid. But by the identity theorem this means that our identity ought to keep holding as we move $z$ smoothly around the branch points. Which means that the identity doesn't really take principal branches into account after all.
We can cheat this argument by picking uncommon ways to write down $f$ and $g$ (or by choosing a sufficiently uninteresting identity such as $\operatorname{Log}z = \operatorname{Log}z$) -- but generally that would mean that using the identity will itself require the same kind of painstaking analysis you're hoping to get out of.
