  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 a
solution 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 field
which, 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 classified
real $2$ and $3-$dimensional supeflows over $\mathbb{R}$.
  In this third part we classify all $2-$dimensional complex superflows; that
is, whose group of symmetries are finite subgroups of $U(2)$. This includes
both irreducible and reducible superflows.
