  A chiral random matrix model with complex eigenvalues is solved as an
effective model for QCD with non-vanishing chemical potential. We derive new
matrix model correlation functions which predict the local fluctuations of
complex Dirac operator eigenvalues at zero virtuality. The parameter which
measures the non-Hermiticity of the Dirac matrix is identified with the
chemical potential. In the phase with broken chiral symmetry all spectral
correlations of the Dirac eigenvalues are calculated as functions of quark
flavors and chemical potential. The derivation uses the orthogonality of the
Laguerre polynomials in the complex plane. Explicit results are given for any
finite matrix size N as well in the large-N limit for weak and strong
non-Hermiticity.
