Although presently controversial, some recent detailed studies of the relative positions of heavy element optical transitions and comparison with present day wavelengths suggest that the fine structure constant, &alpha&equiv e²/hbar c, may have evolved with time. Due to the different &alpha -dependences of the Coulombic and magnetic moment interactions, comparison of atomic optical, HI 21-cm and molecular millimetre transitions can yield at least an order of magnitude in precision over the purely optical results. This, however, is severely limited by the low number of redshifted systems exhibiting HI and optical/rotational absorption currently known. Here we discuss how, with its unprecedented sensitivity and large tuning range, the Square Kilometre Array (SKA) is expected to significantly increase the number of known high redshift radio absorbers, thus greatly improving measurements of the variation in the fine structure constant and electron-to-proton mass ratio in the early Universe.