What is √i\sqrt{i}?

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

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