We have explored the nature of the extended emission-line region around the z=0.37 quasar 4C 37.43, using extensive ground-based and HST imaging and spectroscopy. The velocity field of the ionized gas shows gradual gradients within components but large jumps between components, with no obvious global organization. The HST [O III] image shows radial linear features on the east side of the QSO that appear to mark the edges of an ionization cone. Concentrating on the bright emission peaks ~4arcsec$ east of the quasar, we find through modeling that we require at least two density regimes contributing significantly to the observed emission-line spectrum: one with a density of ~2 cm^-3, having essentially unity filling factor, and one with a density of ~500 cm^-3, having a very small (~10^-5) filling factor. Because the temperatures of these two components are similar, they cannot be in pressure equilibrium, and there is no obvious source of confinement for the dense regions. We estimate that the dense regions will dissipate on timescales <~10^4 years and therefore need to be continuously regenerated, most likely by shocks. Because we know that some QSOs, at least, begin their lives in conjunction with merger-driven massive starbursts in their host galaxies, an attractive interpretation is that the extended emission region comprises gas that has been expelled as a result of tidal forces during the merger and is now being shocked by the galactic superwind from the starburst. This picture is supported by the observed distribution of the ionized gas, the presence of velocities ranging up to ~700 km s^{-1}, and the existence of at least two QSOs having similarly luminous and complex extended emission regions that are known to have ultra-luminous IR galaxy hosts with current or recent starbursts.
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