Polymer membranes with improved transport properties are required for effective separation of organic mixtures (such as methanol‒hexane system) by combination of pervaporation and azeotropic distillation. The present work is devoted to comparative study of two types of membranes based on poly(amidoimide acid) with 2,2′-biquinoline-6,6′ units in the backbone; the objects were prepared (i) from the initial polymer and (ii) from the polymer-metal complex (with Cu(I)). Thermo-mechanical and mass spectrometric investigations demonstrated good operational properties of the samples. Density measurements and SEM analysis revealed that the structure formed in polymer-metal complex is more compact as compared to that of the pure polymer membrane. Mass transfer processes of methanol and hexane through both kinds of membranes were studied by sorption, desorption and pervaporation tests. The values of equilibrium sorption degree, the Flory–Huggins parameter, and diffusion coefficient were determined for the obtained membranes. The pervaporation data allowed calculating permeability and selectivity of membranes in addition to the flux and the separation factor. The membrane based on polymer-Cu(I) complex allowed separating the methanol‒hexane azeotropic mixture with a separation factor of 980 and pervaporation separation index equal to 66.6; therefore, this process is significantly more effective than separation procedures involving other known membranes.