  Processes related to electronically excited states are central in many areas
of science, however accurately determining excited-state energies remains a
major challenge in theoretical chemistry. Recently, higher energy stationary
states of non-linear methods have themselves been proposed as approximations to
excited states, although the general understanding of the nature of these
solutions remains surprisingly limited. In this Letter, we present an entirely
novel approach for exploring and obtaining excited stationary states by
exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres
on performing analytic continuations of conventional quantum chemistry methods.
Considering Hartree--Fock theory as an example, we analytically continue the
electron-electron interaction to expose a hidden connectivity of multiple
solutions across the complex plane, revealing a close resemblance between
Coulson--Fischer points and non-Hermitian degeneracies. Finally, we demonstrate
how a ground-state wave function can be morphed naturally into an excited-state
wave function by constructing a well-defined complex adiabatic connection.
