If q is the probability of the joint occurrence of two independent and equally likely events, then the probability of each one of them occurring is given by the square root of q. So it can be made to mean something.

Is the square root of a probability also a probability in general?

As in real/complex analysis for given [$]x[$] find (unique or otherwise [$]y[$] such that [$]y^2= x[$]. In this case it it a complex number.

[$]\sqrt[2]{x}[$] is not a fundamental quantity. It is a kind of formalism. It's the same process as defining the integers [$]{...,-2,-1,0,1,2,...}[$] as ordered pairs of the natural numbers [$]{...,0,1,2,...}[$] BTW this was the analogy what Feynman alluded to in his note on negative probability although he did not do the axioms AFAIK.

Very parsimonous. The trick is to get new results with as few axioms as possible and these results should build on previous results and not by pulling them out of a magician's hat. Number systems are built incrementally. Should be same with probabilities IMO.