Although the question does not explicitly say so, it is usually assumed that hyperbolic space consists of the upper half of the complex plane. So in this problem I think it's safe to ignore the lower half of the plane.

To calculate $\displaystyle |z|^2$, you have to take the real part of z and the imaginary part of z, square them both, and then add. If $\displaystyle z = a+ib$ (where a and b are both real) then $\displaystyle z-1 = a+ib-1 = (a-1) + ib$. The real part is $\displaystyle a-1$ and the imaginary part is b. So $\displaystyle |z-1|^2 = (a-1)^2+b^2$.

Good grief, I think you're right.

This is very advanced material, and it looks as though this course assumes a lot of previous experience of working with complex numbers and their geometry.