I took only a first course of complex analysis.
I have learned some multi-valued functions which consistently extend the classical real functions such as Log and Trigonometric functions.
However, I don't see any advantage of considering all the braches. I think it's really convenient to consider $(-\pi,\pi]$ branch so that I can treat those multi-valued function as single-valued functions.
Moreover, it's written in wikipedia that if one knows Riemann Surface, there is no need to separate branches.
Is it okay to just not consider all branches , but to focus on the Principal values?