A frequency analysis of 211 h of high-speed photometric observations obtained in 1981, 1983 and 1992 shows that alpha Cir pulsates in two modes with frequencies nu1 = 2442.023 +/- 0.016 muHz and nu2 = 2439.427 +/- 0.016 muHz, separated by nu1 - nu2 = 2.60 +/- 0.02 muHz. This separation suggests that nu1 is a mode of overtone n almost-equal-to 35-40 and degree l = 0, while nu2 has overtone and degree of n - 1 and l = 2. The principal frequency, nu1, however, is phase-modulated. The modulation period-may be 524.2 d, in which case alpha Cir is a binary star with a reasonable probability that the secondary is of substellar mass. The modulation period may be 0.995 350 d, which is then the rotation period, in which case it can be shown that the principal frequency cannot be represented purely by a radial spherical harmonic pulsation mode. A decomposition of a derived frequency triplet indicates that, for most values of the rotational inclination i and the magnetic obliquity beta, the mode is mostly a radial mode with a smaller dipole distortion. We suggest that the rotational inclination i less-than-or-equal-to 12-degrees and the magnetic obliquity is large, which means that the magnetic field is seen always from an unfavourable aspect. The suggested rotation period, P(rot) = 0.995 350 +/- 0.000 005 d, is so close to one sidereal day that it must be confirmed with multisite observations. If it is not confirmed, then the binary model becomes more attractive.