Package ComplexAnalysis does not appear in the list of Standard Extra Packages, no file ComplexAnalysis.m seems to be in $InstallationDirectory, and the Documentation Center includes no entry for either ComplexAnalysis or BranchCuts!

However, if one evaluates an expression using function BranchCuts in context ComplexAnalysis, then

 ?ComplexAnalysis`BranchCuts

does reveal a usage message.

Very strange!

Very cool! Where/how did you find the function?

How to you determine the branch lines? I believe they are somewhat arbitrary. (Correct me if I'm wrong!) For standard functions in Mathematica the function Help pages generally tell you how the branch lines are chosen. For arbitrary functions you are relatively free to pick the paths yourself.

If the function you are looking at has a small finite number of critical points or branch points then it can be represented on a Riemann surface and has no branch lines at all. My Presentations application has Multivalues routines that allows one to plot a function on a Riemann surface. The function value is represented by a vector attached to a Locator point. As you drag the locator around the vector will change but it is always smooth and single valued. For the Sqrt[z] you can drag it around the origin without any discontinuities but you have to go around twice to return the same value. You have traversed different portions of the Riemann surface.