Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is no single constant bounding the quantifier depth of the defining formulae.
Q1)When does $Th(\mathcal{M})$ admit quantifier elimination?
Q2) When is $Th(\mathcal{M})$ decidable?