  Akin to the idea of complete sets of Mutually Unbiased Bases for prime
dimensional Hilbert spaces, $\mathcal{H}_d$, we study its analogue for a $d$
dimensional subspace of $M (d,\mathbb{C})$, i.e. Mutually Unbiased Unitary
Bases (MUUBs) comprising of unitary operators. We note an obvious isomorphism
between the vector spaces and beyond that, we define a relevant monoid
structure for $\mathcal{H}_d$ isomorphic to one for the subspace of $M
(d,\mathbb{C})$. This provides us not only with the maximal number of such
MUUBs, but also a recipe for its construction.
