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 K$_2$CuSO$_4$Cl$_2$ 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 non-universal spectral features of the data is a novel thermally activated longitudinal mode that remains underdamped across most of the Brillouin zone.
