I feel like you're being technically correct but in a kind of useless way. Yes, nothing stops you from defining the square root function such that it returns a

I feel like you're being technically correct but in a kind of useless way. Yes, nothing stops you from defining the square root function such that it returns an element of R^2. It's perfectly legal mathematics, functions can map arbitrary domains to arbitrary codomains after all. But I've never seen it used anywhere. Maybe there is an area where such a definition makes sense... but regular real (or complex) analysis doesn't really normally do such a kind of thing. The problem is that you want your typical functions like square root (and logarithms, which in the complex case also aren't unique) to be functions R -> R (or C -> C). Otherwise you can't compose them and then everything is kinda weird. In particular, you don't get the vector space structure that the article is all about. When working in a structure, one often doesn't want to leave that structure and go to another one, but keep the same structure (or at least one that is related in an interesting way, such as a quotient structure). That just leads to more interesting mathematics.