  Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only
separable C*-algebras that admit quantifier elimination in continuous logic are
$\mathbb{C},$ $\mathbb{C}^2,$ $M_2(\mathbb{C}),$ and the continuous functions
on the Cantor set. We show that, among finite dimensional C*-algebras,
quantifier elimination does hold if the language is expanded to include two new
predicate symbols: One for minimal projections, and one for pairs of unitarily
conjugate projections. Both of these predicates are definable, but not
quantifier-free definable, in the usual language of C*-algebras. We also show
that adding just the predicate for minimal projections is sufficient in the
case of full matrix algebras, but that in general both new predicate symbols
are required.
