The present paper deals with mutually unbiased bases for systems of qudits in d dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of 1 + p mutually unbiased bases is given for d = p where p is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group SU(2). A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case when d = pe (e >= 2), corresponding to the power of a prime integer, is briefly examined. Finally, complete sets of mutually unbiased bases are analysed through a Lie algebraic approach.