Accretion at super-Eddington rates is expected to be accompanied by strong outflows. Such outflows are observed in Galactic X-ray binaries and extragalactic Ultra-luminous X-ray sources (ULXs). However, due to their large source distances, ULX outflows are challenging to detect and study in detail. Galactic neutron stars accreting from a Be-star companion at super-Eddington rates show many similarities to ULX pulsars, and therefore offer an alternative approach to study outflows in this accretion regime. Here, we present Chandra high-resolution spectroscopy of such a super-Eddington accreting neutron star, Swift J0243.6+6124, to search for wind outflow signatures during the peak of its 2017/2018 giant outburst. We detect narrow emission features at rest from Ne, Mg, S, Si, and Fe. In addition, we detect a collection of absorption features which can be identified in two ways: either as all Fe transitions at rest (with a possible contribution from Mg), or a combination of three blue-shifted Ne and Mg lines at ∼ 0.22c, while the remaining lines are at rest. The second scenario would imply an outflow with a velocity similar to those seen in ULXs, including the ULX pulsar NGC 300 ULX-1. This result would also imply that Swift J0243.6+6124 launches both a jet, detected in radio and reported previously, and an ultra-fast wind outflow simultaneously at super-Eddington accretion rates.