A chiral random matrix model with complex eigenvalues is solved as an effective model for QCD with nonvanishing chemical potential. The new correlation functions derived from it are conjectured to predict the local fluctuations of complex Dirac operator eigenvalues at zero virtuality. The parameter measuring the non-Hermiticity of the random matrix is related to the chemical potential. In the phase with broken chiral symmetry all spectral correlations are calculated for finite matrix size N and in the large-N limit at weak and strong non-Hermiticity. The derivation uses the orthogonality of the Laguerre polynomials in the complex plane.