Idk, to me it feels much much better than just picking one root when defining the inverse function.
This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).
Well, yeah, the alternative is also bad.
But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.