Effect of multiplication by square root of i The problem requires an explanation of the 'effect' of the multiplication of a complex number, $a + bi$, by $\frac{1+i}{\sqrt{2}}$ i.e. the positive square root of $i$. The question was ambiguous with regard to what, precisely, constitutes an 'explanation', but I suspect that the multiplication entails some distinct geometric transformation. I proceeded this way: $$\begin{align} (a+bi)\sqrt{i} &= \frac{a+ai+bi+bi^2}{\sqrt{2}}\\\ &= \frac{a-b}{\sqrt{2}} + \frac{a+b}{\sqrt{2}}i \end{align}$$ I cannot deduce anything further from this result, however, and a quite comprehensive scour of the internet did not yield anything relevant. I would appreciate any opinion on the nature of the 'effect' that the question refers to.

It's a rotation to 45 degrees i.e. to $\frac{\pi}{4}$.

Any multiplication by a complex number $z = r(cos\phi + i.sin\phi)$ is a rotation to an angle of size $\phi$ and then a multiplication of the original number's norm to $r$ (i.e. to $z$'s norm).
In your case $r=1$ so you're left only with the rotation.

I read a good book about this 20 years ago but it's in Bulgarian only.
Try to google for "complex numbers in geometry".