  We explore the non-Hermitian extension of quantum chemistry in the complex
plane and its link with perturbation theory. We observe that the physics of a
quantum system is intimately connected to the position of complex-valued energy
singularities, known as exceptional points. After presenting the fundamental
concepts of non-Hermitian quantum chemistry in the complex plane, including the
mean-field Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation
theory, we provide a historical overview of the various research activities
that have been performed on the physics of singularities. In particular, we
highlight seminal work on the convergence behaviour of perturbative series
obtained within M{\o}ller--Plesset perturbation theory, and its links with
quantum phase transitions. We also discuss several resummation techniques (such
as Pad\'e and quadratic approximants) that can improve the overall accuracy of
the M{\o}ller--Plesset perturbative series in both convergent and divergent
cases. Each of these points is illustrated using the Hubbard dimer at half
filling, which proves to be a versatile model for understanding the subtlety of
analytically-continued perturbation theory in the complex plane.
