According to wikipedia's page on signed zeroes, we agree that $\sqrt{-0} = -0$.
I would always have guessed that it would be $0i$, as $(0i)^2 = 0^2*i^2 = 0 * (-1) = -0$.
I know that my own reasoning is not correct, as $0i$ is a multiplication that would simply result in $0$, but why have we resorted to such an un-intuitive rule instead of creating a symbol to denote an "irrational zero"?
I think when the floating point type was invented the easiest thing to do for having positive and negative numbers was just introducing an additional bit that represents the sign. This now results in the problem that you have two representations for the number 0. Of course $0=-0$ in a mathematical sense.
But now you had to 'invent' rules on how to handle calculations and for making an universal convention like those IEEE standards the rules should be defined for every case possible. And $\sqrt{-0} = -0$ seems to be a pretty intuitive solution. Keep in mind that this is just the rule how to implement these standards, it has nothing to do with math since in math $\sqrt{-0} = \sqrt{0} = 0 = -0$.
Introducing a complex zero into the 'float' definition would require introducing complex numbers which wouldn't be a great idea since I think more thant 90 percent of all calculations that require floats do not need complex numbers but are inherently just real numbers so you would waste a lot of memory if you had to save the imaginary part (of value 0 most of the time) each time.