In a number system with multiple nonreals, p2=1, q2=-1, and/or r2=0 should have multiple solutions, right? For example, in the quaternions, q could be i, j, or k. Is it multivalued then?
And what happens when you're in a mixed signature? If I'm not mistaken, any combination of positive roots could be a part of q, and any combination of positive or negative roots could be a part of r, as long as q contains one negative root and r contains one root of zero.
i2 = (iJ)2 = ... = -1 ε2 = (εi)2 = (εiJ)2 = ... = 0
I'm being very careful not to write $\mathbb"\sqrt-1 = i"$ and so on because I know they're not supposed to be identically equal, or $\mathbb(ij=-1$).
But that just begs the question, what is sqrt(-1), etc. in quaternions and beyond? Is it considered multivalued or undefined or what?
Personally I would lean toward a separate variant of sqrt derived from each generator, so sqrti(-1) is unambiguously i via i2=-1. The general sqrt(-1) would be undefined, since none of the generators are identically equal to it. But I imagine I'm completely alone on this one.
Sorry about my formatting and lack of math background/vocabulary. Also this will likely be marked a duplicate of another question, but I couldn't find the right one.