Are multi-valued functions a rigorous concept or simply a conversational shorthand?
Solution 1:
Multivalued analytic functions can be made (and have been made) a rigorous notion. This notion is sometimes useful. But modern textbooks prefer not to use it, because it is hard to deal with rigorously. (What is a sum or product of multivalued functions?)
There are several substitutes:
To use only single valued branches. For this you need to restrict the region (usually by making some "branch cuts"). This is the way most elementary textbook take.
To use sheafs (which can be considered a rigorous framework into which multivalued functions fit, though there are alternative approaches). This is the approach used in the standard graduate textbook of Ahlfors.
To translate everything to the language of functions on Riemann surfaces. This is perhaps the most useful approach, at least in one complex variable, which was proposed by Riemann.