Your second expression is incorrect and unnecessary, since is a single point on the complex plane.
However, it does have 2 square roots.
De Moivre's Theorem gives
You will have to use the 2 values of ,......
Using Pythagoras' theorem, the length of is
Therefore the length of the square root is 2.
The argument of is
To find solutions for from , realise that in squaring , the double angle of is
We therefore also take a 2nd angle for , the angle
These angles are
so the square root of at angle is
The negative square root is
These complex numbers differ by radians.
The reason we take 2 angles is because in the complex world, the angles
and are the same, as they reference the exact same position.
When a complex number is squared, it's angle doubles,
so a complex number with an angle between and
when squared has an angle between and .