Constraining the difference in convective blueshift between the components of alpha Centauri with precise radial velocities

New radial velocities of alpha Cen A & B obtained in the framework the Anglo-Australian Planet Search programme as well as in the CORALIE programme are added to those by Endl et al. (\cite{Endl-2001:a}) to improve the precision of the orbital parameters. The resulting masses are 1.105+/-0.0070 M_sun and 0.934+/-0.0061 M_sun for A and B respectively. The factors limiting how accurately these masses can be derived from a combined visual-spectroscopic solution are investigated. The total effect of the convective blueshift and the gravitational redshift is also investigated and estimated to differ by 215 +/- 8 m s^-1 between the components. This suggests that the difference in convective blueshift between the components is much smaller than predicted from current hydrodynamical model atmosphere calculations.