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Let $X\in\mathbb{R}^{n}$ or $\mathbb{C}^{n}$. For$\phi:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ (respectively,$\phi:\mathbb{C}^{n}\mapsto\mathbb{C}^{n}$) and $t\in\mathbb{R}$ (respectively,$\mathbb{C}$), we put $\phi^{t}=t^{-1}\phi(Xt)$. A projective flow is asolution to the projective translation equation$\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$ or $\mathbb{C}$.  The projective superflow is a projective flow with a rational vector fieldwhich, among projective flows with a given symmetry, is, up to a homothety,unique and optimal. In the first and the second part of this work we classifiedreal $2$ and $3-$dimensional supeflows over $\mathbb{R}$.  In this third part we classify all $2-$dimensional complex superflows; thatis, whose group of symmetries are finite subgroups of $U(2)$. This includesboth irreducible and reducible superflows.

Projective superflows. III. Finite subgroups of $U(2)$