We review the possible formation and orbital stability of Earth-mass or super Earth-mass planets around either of the stars Alpha Centauri A or B and describe a program at Mt John University Observatory using the Doppler method that aims to detect such planets. From New Zealand, we are able to observe the Alpha Centauri system year-round. This is critical in order to acquire data of sufficient quantity and phase coverage to detect the orbit of a terrestrial-mass planet in the habitable zone. Our observations are being made at high resolution (R = 70,000) and high signal-to-noise with the Hercules vacuum echelle spectrograph attached to the 1-m McLellan telescope by a 25-m long optical fibre and using an iodine cell. We discuss the velocity precision and instrumental stability required for success and outline the progress of the observations so far. At present we are collecting about 10,000 observations of each star, A and B, per year with a typical precision of 2.5 m/s per observation.