I'm interested in understanding the Riemann surface associated with the mapping of the complex number $z$ to the complex number $w$ where $z = (w + 1)(w + i) $.
I thought there would be two "sheets" and that $w = -1$ and $w = -i$ would be branch points.
I attempted to code a representation this complex transformation in two different ways, which I’ve linked to below. In the first, a square on the left represents the patch of the complex plane centered on the origin. It is colored according to phase. The four squares to the right show where these colored points get mapped to. In each case the center of the square is the origin and the dimensions are -2 to 2, -2i to 2i. The w = -1 and w = -I are marked in one square with black dots but they don’t seem significant. Shouldn’t they be the center of rainbow swirls because they correspond to zero in the domain plane?
The second representation has the x-y plane represent the complex domain. The height of the surface represents the real portion of w and the color represents the imaginary portion of w. (The switch button reverses this). Sliders let one change the real and imaginary parts of $B$ and $C$, where $z = w^2 + Bw + C$. The initial values of B and C are $1+i$ and $i$, respectively. You can manipulate your view with the mouse. This graph also does not seem to show a branch point where I’d expect.
Is there a flaw in the code of each?