Inelastic neutron-scattering and finite-temperature density matrix renormalization group (DMRG) calculations are used to investigate the spin excitation spectrum of the S = 1/2 Heisenberg spin chain compound K2 CuSO4 Cl2 at several temperatures in a magnetic field near saturation. Critical correlations characteristic of the predicted z = 2, d = 1 quantum phase transition occurring at saturation are shown to be consistent with the observed neutron spectra. The data is well described with a scaling function computed using a free fermion description of the spins, valid close to saturation, and the corresponding scaling limits. One of the most prominent nonuniversal spectral features of the data is a novel thermally activated longitudinal mode that remains underdamped across most of the Brillouin zone.