Following the discovery of X-ray absorption in a high-velocity outflow from the bright quasar PG 1211 + 143 we have searched for similar features in XMM–Newton archival data of a second (high accretion rate) quasar PG 0844+349. Evidence is found for several faint absorption lines in both the EPIC and RGS spectra, whose most likely identification with resonance transitions in H-like Fe, S and Ne implies an origin in highly ionized matter with an outflow velocity of order ∼0.2c. The line equivalent widths require a line-of-sight column density of NH∼ 4 × 1023 cm−2, at an ionization parameter of log ξ∼ 3.7. Assuming a radial outflow being driven by radiation pressure from the inner accretion disc, as suggested previously for PG 1211 + 143, the flow in PG 0844+349 is also likely to be optically thick, in this case within ∼25 Schwarzschild radii. Our analysis suggests that a high-velocity, highly ionized outflow is likely to be a significant component in the mass and energy budgets of active galactic nuclei accreting at or above the Eddington rate.