Problem with plotting complex valued function from (D,D) to D

Question

I'm having problem in plotting a complex valued function $D \times D$ to $D$. Defined by $$ f(z_1, z_2) = \max(z_1, z_2) = $$ if $|z_1| > |z_2|$ $$ \max(z_1, z_2) = z_1 $$
else if $|z_1| < |z_2|$ $$ \max(z_1, z_2) = z_2 $$
else if $|z_1| = |z_2| $ and $\arg(z_1) > \arg(z_2)$ $$ \max(z_1, z_2) = z_1 $$ else $$ \max(z_1, z_2) = z_2 $$ .$D = \{z\in C : |z| <= 1\}$ i.e. closed unit disk centered at the origin of the complex plane. I know several plotting or visualization techniques $C \longrightarrow C$, But could not find out any such $C \times C \longrightarrow C$. If every details of the mapping cannot be covered to a single image i.e. total visualization is done by multiple images, still it is okay for me.