Using computational approaches we predict the microstructure of high-performance alkaline polymer, quaternary ammonium polysulfone hydroxide (QAPS-OH) membranes, dry and with similar to 14 wt% water uptake. The microstructure can be described as a hydrophobic polymer backbone penetrated by a network of three-dimensional interlinked hydrophilic channels of different diameters. Mobile OH-anions are distributed inside the channels. OH diffusion coefficients and corresponding activation energy were calculated from our molecular dynamics simulations of the QAPS-OH membrane at different temperatures. The predicted values are consistent with available experimental data. Possible mechanisms of the OH-anion diffusion have been discussed.