If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?   

Is it used or considered often in mathematics? How is it notated?

Let $z=(a+bi)$ be a complex number which is a square root of $i$, that is 
$$i=z^2=(a^2-b^2)+2abi.$$ 
Equating real and imaginary parts we have,
$$a^2-b^2=0, 2ab=1$$

The two real solutions to this pair of equations are
$a={1 \over \sqrt{2}},b={1 \over \sqrt{2}}$ and $a=-{1 \over \sqrt{2}},b=-{1 \over \sqrt{2}}$. The two square roots of $i$ therefore are 


  $$\pm {1 \over \sqrt{2}} (1+i)$$

