There's a lot of great information here about understanding the branch cuts and branch points of functions of the form ( for example ) $(z^3+1)^{1/2}$, sums of simple roots and products thereof.
However, I am not entirely sure how to deal with more complicated examples, such as
$g(z)=(z+ \sqrt{z^2 - 3})^{1/3}$, or even worse iterations (for example, some rational function f[g(z)]).
With the simpler examples that I mentioned at the top, it is easy to get intuition for example by plotting in mathematica for example for $(z+1)^{1/3}$ we can see nicely that one needs three different sheets to obtain a smooth function over a larger domain than the complex plane in the plot
$(z+1)^{1/3}$ for $z=r Exp(I y)$ while varying $r$ from $0$ to $10$ and $y$ from $0$ to $6\pi$" />
When one tries a similar thing for $g(z)=(z+ \sqrt{z^2 - 3})^{1/3}$, I cannot understand the output by mathematica, seemingly not following the correct root branches. How would one do the proper analysis 'by hand' so as to predict this structure?