and a quantum computer could have a number be simultaneously 1 and -1 due to superposition would that allow the square root to be solved by a quantum representation that is a combination of both?
I don't think that has anything to do with quantum or not, and I'm not sure what there is to "solve" about the sqrt(-1). We know what it is, it's just called i in the complex numbers (by definition). There's not another level to go down and explore. (in some other number systems, like the reals, sqrt(-1) just doesn't exist because no number times itself is -1).
1 = (1, 0)
i = (0, 1)
2 + 3i = (2, 3)
and so on.
https://www.obliquity.com/computer/fortran/datatype.html
Fortran IV, did, too:
http://www.math-cs.gordon.edu/courses/cs323/FORTRAN/fortran....
...in 1957.