High-redshift quasars can be used to trace the early growth of massive galaxies and may be triggered by galaxy–galaxy interactions. We present Multi-Object Spectroscopic Explorer (MUSE) science verification data on one such interacting system consisting of the well-studied z = 3.2 PKS1614+051 quasar, its AGN companion galaxy and bridge of material radiating in Ly α between the quasar and its companion. We find a total of four companion galaxies (at least two galaxies are new discoveries), three of which reside within the likely virial radius of the quasar host, suggesting that the system will evolve into a massive elliptical galaxy by the present day. The MUSE data are of sufficient quality to split the extended Ly α emission line into narrow velocity channels. In these the gas can be seen extending towards each of the three neighbouring galaxies suggesting that the emission-line gas originates in a gravitational interaction between the galaxies and the quasar host. The photoionization source of this gas is less clear but is probably dominated by the two AGN. The quasar's Ly α emission spectrum is double peaked, likely due to absorbing neutral material at the quasar's systemic redshift with a low column density as no damping wings are present. The spectral profiles of the AGN and bridge's Ly α emission are also consistent with absorption at the same redshift indicating that this neutral material may extend over >50 kpc. The fact that the neutral material is seen in the line of sight to the quasar and transverse to it, and the fact that we see the quasar and it also illuminates the emission-line bridge, suggests that the quasar radiates isotropically and any obscuring torus is small. These results demonstrate the power of MUSE for investigating the dynamics of interacting systems at high redshift.