AX J1745.6-2901 is a high-inclination (eclipsing) neutron star Low Mass X-ray Binary (LMXB) located less than ~1.5 arcmin from Sgr A*. Ongoing monitoring campaigns have targeted Sgr A* frequently and these observations also cover AX J1745.6-2901. We present here an X-ray analysis of AX J1745.6-2901 using a large dataset of 38 XMM-Newton observations, including eleven which caught AX J1745.6-2901 in outburst. Fe K absorption is clearly seen when AX J1745.6-2901 is in the soft state, but disappears during the hard state. The variability of these absorption features does not appear to be due to changes in the ionizing continuum. The small Kalpha/Kbeta ratio of the equivalent widths of the Fe xxv and Fe xxvi lines suggests that the column densities and turbulent velocities of the absorbing ionised plasma are in excess of N_H ~ 10^23 cm^-2 and v_turb >~ 500 km s^-1. These findings strongly support a connection between the wind (Fe K absorber) and the accretion state of the binary. These results reveal strong similarities between AX J1745.6-2901 and the eclipsing neutron star LMXB, EXO 0748-676, as well as with high-inclination black hole binaries, where winds (traced by the same Fe K absorption features) are observed only during the accretion-disc-dominated soft states, and disappear during the hard states characterised by jet emission.