We analyze the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero chemical potential when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. A comparison with predictions from random matrix theory is presented. We further provide first evidence that matrix models describe the low lying complex Dirac eigenvalues in a theory with dynamical fermions at nonzero chemical potential. Lattice data for two-color QCD with staggered fermions are compared to detailed analytical results from matrix models in the corresponding symmetry class, the complex chiral symplectic ensemble. They confirm the predicted dependence on chemical potential, quark mass and volume.