In most quantum mechanical systems, the Hamiltonian under consideration is Hermitian. However, physically meaningful extensions of quantum mechanics to non-Hermitian systems have existed since the 60s. More recently, many advances have been made with PT-symmetric systems, in which Hermiticity is replaced with a parity plus time-reversal (PT) symmetry. In this talk, a one-dimensional discrete non-Hermitian quantum system with a special exact solution will be discussed. This solution takes the form of free parafermions, which generalises the notion of free fermions found in the solution of many spin chains. Some unusual properties of this model will be presented, including an anomalous bulk energy, and divergent expectation values of some simple operators.