In this paper we address the blind separation of an instantaneous mixture of statistically independent complex non-circular signals. We show that in limited cases, by exploiting all the second-order information, the mixing matrix can be estimated up to a real orthogonal factor. This is based on a link with the Takagi factorization, for the computation of which we derive a Jacobi-type algorithm. We prove that, under more common assumptions, after a classical prewhitening, the remaining unitary factor can be found via a simultaneous Takagi factorization / Hermitian Eigenvalue Decomposition (EVD). We also describe a variant in which no hard prewhitening is carried out. In addition, we pay some attention to the issue of dimensionality reduction, in the case where there are fewer sources than sensors.