Taking the square root of an imaginary number We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota).
The question is, what would happen if take square root of a positive imaginary number? Will the dimension split up again, or will it stay in level-1 imaginary dimension?


Basically instead of degrees and radians, on a complex plane we can use multiplication to express rotations. Multiplying by $1$ takes you take to the same place, so it's the equivalent of 360 degrees. Multiplying by $-1$ is the same as 180 degrees (because if you do that two times it's the equivalent of multiplying by $1$). Multiplying by $i$ is the same as a 90 degree rotation and multiplying by  $\sqrt{i}$ is the same as rotating 45 degrees (because if you multiply by $\sqrt{i}$ eight times it's the equivalent of a $360$ rotation). Looking at the value onto the imaginary unit circle of 45 degrees can give you the value  of $\sqrt{i}$. However you also need to consider the negative value is also a solution when dealing with any even roots.
In general multiplication by $\sqrt[n]{i}$ is the same as a $90/n$ degree rotation so you can find any root of  $i$ if you want. 
We use multiplication to represent rotations on a complex plane because we don't graph $x$ vs $y$, rather we graph one number as a whole.
