Mutually Unbiased Bases (MUBs) are closely connected with quantum physics and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for \(\mathbb {C}^n\) by studying real points of a certain affine algebraic variety. This variety comes from the relations that determine the extendability of a system of MUBs. Finally, we show that some part of this variety gives rise to complete intersection domains. Further, we show that there is a one-to-one correspondence between MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in \(\mathcal {M}_n({\mathbb {C}})\) . It means that for m MUBs in \(\mathbb {C}^n\) , there are m commuting classes each consisting n commuting orthogonal normal matrices and the existence of maximal commuting basis for \(\mathcal {M}_n({\mathbb {C}})\) ensures the complete set of MUBs in \(\mathcal {M}_n({\mathbb {C}})\) .
