Pseudo-unitary non-self-dual fusion categories of rank 4

A fusion category of rank 4 has either four self-dual simple objects or exactly two self-dual simple objects. We study fusion categories of rank 4 with exactly two self-dual simple objects, giving nearly a complete classification of those based rings that admit pseudo-unitary categorification. More precisely, we show that if CC is such a fusion category, then its Grothendieck ring K(C)K(C) must be one of seven based rings, six of which have known categorifications. In doing so, we classify all based rings associated with near-group categories of the group Z/3ZZ/3Z.