A quantum critical point (QCP) is a singularity in the phase diagram arising due to 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 the critical quantum and thermal fluctuations near such a point. The direct and continuous control of these fluctuations has historically been difficult to realize, but has been achieved recently in a high-pressure, high-resolution neutron scattering experiment on the quantum dimer material TlCuCl_3. Measurements of the magnetic excitation spectrum across the entire quantum critical phase diagram, in pressure and temperature, demonstrate a number of remarkable properties arising at the interface between quantum and classical physics. Quantum and thermal fluctuations have very similar effects in melting the magnetically ordered phase and in opening excitation gaps. In the QC regime there is robust ω/T scaling of the energies and Γ/T scaling of the widths of well-defined but critically damped excitations. This QC scaling crosses over to a classical critical form in a narrow region around the antiferromagnetic transition line T_N(p); the static and dynamic scaling properties of these two types of criticality indicate that quantum and thermal fluctuations can behave largely independently near a QCP. The critically damped longitudinal, or Higgs, mode of the ordered phase is exquisitely sensitive to thermal fluctuations and becomes overdamped in the classical regime.