Physicist: The beauty of complex numbers (numbers that involve ) is that the answer to this question is a surprisingly resounding: nopers.
The one thing that needs to be known about is that, by definition, . Other than that it behaves like any other number or variable. It turns out that the square root is . You can check this the same way that you can check that 2 is the square root of 4: you square it.
And like any other square root, the negative, , is also a solution. So, does have a square root, and it’s not even that hard to find it. No new “super-imaginary” numbers need to be invented.
This isn’t a coincidence. The complex numbers are “algebraically closed“, which means that no matter how weird a polynomial is, it’s roots are always complex numbers. The square roots of , for example, are the solutions of the polynomial . So, any cube root, any Nth root, any power, any combination, any whatever of any complex number: still a complex number.