A quantum critical point (QCP) is a singularity in the phase diagram arising because of quantum mechanical fluctuations. The exotic properties of some of the most enigmatic physical systems, including unconventional metals and superconductors, quantum magnets, and ultracold atomic condensates, have been related to the importance of critical quantum and thermal fluctuations near such a point. However, direct and continuous control of these fluctuations has been difficult to realize, and complete thermodynamic and spectroscopic information is required to disentangle the effects of quantum and classical physics around a QCP. This control has been achieved in a high-pressure, high-resolution neutron scattering experiment on the quantum dimer material TlCuCl3. Measurements of the magnetic excitation spectrum across the entire quantum critical phase diagram illustrate the similarities between quantum and thermal melting of magnetic order. The intrinsically critical nature of the unconventional longitudinal ("Higgs") mode of the ordered phase may be demonstrated by damping it thermally. Two types of criticality develop around the QCP, namely quantum and classical, and the static and dynamic scaling properties of each regime suggest that quantum and thermal fluctuations can behave largely independently in three spatial dimensions.