Alpha Centauri is a triple stellar system, and it contains the closest star to Earth (Proxima Centauri). Over the last decades, the stars in Alpha Cen and their orbits have been investigated in great detail. However, the possible scenarios for planet formation and evolution in this triple stellar system remain to be explored further. First, we present a 3D hydrodynamical simulation of the circumstellar discs in the binary Alpha Cen AB. Then, we compute stability maps for the planets within Alpha Cen obtained through N-body integrations. Last, we estimate the radial velocity (RV) signals of such planets. We find that the circumstellar discs within the binary cannot exceed 3 au in radius and that the available dust mass to form planets is about 30 M⊕. Planets around A and B are stable if their semimajor axes are below 3 au, while those around C are stable and remain unperturbed by the binary AB. For rocky planets, the planetary mass has only a mild effect on the stability. Therefore, Alpha Cen could have formed and hosted rocky planets around each star, which may be detected with RV methods in the future. The exoplanetary hunt in this triple stellar system must continue.