How would yo make a branch cut to define a single-valued branch of the function log(z - 1 + i)? The question says it all. I understand the concept of branch cuts, but I have not quite yet figured out how to find branch cuts of functions. If I am not mistaken, the branch cut of logz is πi + 2πz for any integer z.

There are infinite different branch cuts that satisfy this, but for one, you could do $1-i+z$ for $\\{{z\in \mathbb{R}}|z>0\\}$.