  We calculate the expectation value of an arbitrary product of characteristic
polynomials of complex random matrices and their hermitian conjugates. Using
the technique of orthogonal polynomials in the complex plane our result can be
written in terms of a determinant containing these polynomials and their
kernel. It generalizes the known expression for hermitian matrices and it also
provides a generalization of the Christoffel formula to the complex plane. The
derivation we present holds for complex matrix models with a general weight
function at finite-N, where N is the size of the matrix. We give some explicit
examples at finite-N for specific weight functions. The characteristic
polynomials in the large-N limit at weak and strong non-hermiticity follow
easily and they are universal in the weak limit. We also comment on the issue
of the BMN large-N limit.
