I have a complex polynomial $f(z)$ and I would like to plot a 3D graph that takes in $x$ and $y$ (as the real/imaginary parts) and outputs the modulus of the result. How can I write, for example $z=1+(x+iy)+(x+iy)^2$ without using $i$?
The 3D graphing program I have will only accept functions of the form $\phi: R^2 \to R$.
Define (or do it a priori on paper) the function $\Psi:\mathbb{R}^2\rightarrow\mathbb{C}$ by sending $(x,y)$ into $x+iy$. This is a homeomorphism so there's no data lost, and obviously $\Psi^{-1}$ sends $x+iy$ into $(x,y)$. Now work with the function $\Psi\circ f\circ \Psi^{-1}$, which is a function from $\mathbb{R}^2$ to $\mathbb{C}$. Now define the absolute value function normally for it and plot.