I get that in complex analysis there’s a branch cut along the negative real axis for Log z, and you can define this branch anywhere where you want it to be, but has anyone ever found a way of conceptualizing the complex plane so that you don’t need a branch at all? I know most engineering applications never had a problem just defining where the branch is, but has anyone ever heard of someone being able to not have a branch cut at all?
I don’t think so. Riemann surfaces are another way to look at multivalue functions. (Set of nontrivial discrete valuations on function field.)