The Hamiltonian of an open quantum system is non-Hermitian. Its eigenvalues and eigenfunctions are complex and energy dependent. They determine the spectroscopic properties of the system. The eigenvalues may cross in the complex energy plane. The crossing points are branch points that separate the scenario with avoided level crossings from that without any crossing in the complex energy plane. Mathematically, the first case is characterized by level repulsion, the second one by widths bifurcation. The topology of the branch points is different from that of diabolic points: the geometric phase is twice the Berry phase. Physically, the branch points in the complex energy plane cause some stabilization of the system and reduce the phase rigidity of the scattering wave function.