I explained a little bit about complex numbers a while back and why they are plotted in 2d.

After reading this, how about we calculate the square root of i? Mentally of course.

Here we go. The “square root of i” is a process, when executed 4 times equals -1:

x^4=-1.

In a geometric plane, this is easy: it is 1, rotated by 45 degrees! Do this rotation 4 times and we go from 1 to -1.

From the Pythagorean theorem we know that cos(45) = \sqrt{\frac{1}{2}}.

x=y, so if we create a complex number from this x and y, we get:

\sqrt{\frac{1}{2}} + \sqrt{\frac{1}{2}}i

This is the square root of i. Wolfram Alpha confirms our answer.