We present a fully self-consistent, line-by-line differential abundance analysis of $alpha$ Cen AB based on high-quality HARPS data. Various line lists are used and analysis strategies implemented to improve the reliability of the results. Abundances of 21 species with a typical precision of 0.02-0.03 dex are reported. We find that the chemical composition of the two stars is not scaled solar (e.g. Na and Ni excess, depletion of neutron-capture elements), but that their patterns are strikingly similar, with a mean abundance difference (A - B) with respect to hydrogen of -0.01$pm$0.04 dex. Much of the scatter may be ascribed to physical effects that are not fully removed through a differential analysis because of the mismatch in parameters between the two components. We derive an age for the system from abundance indicators (e.g. [Y/Mg] and [Y/Al]) that is slightly larger than solar and in agreement with most asteroseismic results. Assuming coeval formation for the three components belonging to the system, this implies an age of about $sim$6 Gyrs for the M dwarf hosting the terrestrial planet Proxima Cen b. After correction for Galactic chemical evolution effects, we find a trend between the abundance ratios and condensation temperature in $alpha$ Cen A akin to that of the Sun. However, taking this finding as evidence for the sequestration of rocky material locked up in planets may be premature given that a clear link between the two phenomena remains to be established. The similarity between the abundance pattern of the binary components argues against the swallowing of a massive planet by one of the stars after the convective zones have shrunk to their present-day sizes.
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