For an arbitrary complex number, a + bi, we can plot this as a vector on a two axis scale (x axis real and y axis imaginary). We can also convert any complex number to the form z = r e^(i θ). In other words, draw the complex number vector as an angle and a magnitude, in polar form.
So any number can be drawn as a vector onto the complex plane. Multiplying two vectors together causes the angles to add and the magnitude to multiply — creating a rotated and extended (or shrunken) new vector as the product.
So what’s the square root of a complex number z? It’s the vector a that, when multiplied by itself, winds up with z.
When z has no imaginary component, it lies flat on the x axis; its magnitude is z and the angle is 0. What’s its square root?
There are two. |a|e^(i 0) and |a|e^(i 180). 180 degrees multiplied together rotates the vector back to 0. And the e^(i 180), in radians, is e^(i pi), or negative one.
So putting it together: there are only two solutions, with the same magnitude, and different signs.