  Under very strong axioms, there is precisely one real noncommutative geometry
between the classical one and the free one, namely the half-classical one,
coming from the relations $abc=cba$. We discuss here the complex analogues of
this geometry, notably with a study of the geometry coming from the commutation
relations between all the variables $\{ab^*,a^*b\}$, that we believe to be the
"correct" one.
