  We show that, in a highest weight category with duality, the endomorphism
algebra of a tilting object is naturally a cellular algebra. Our proof
generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer. This
result raises the question of whether all cellular algebras can be realized in
this way. The construction also works without the presence of a duality and
yields standard bases, in the sense of Du and Rui, which have similar
combinatorial features to cellular bases. As an application, we obtain standard
bases -- and thus a general theory of "cell modules" -- for Hecke algebras
associated to finite complex reflection groups (as introduced by Brou\'e,
Malle, and Rouquier) via category $\mathcal{O}$ of the rational Cherednik
algebra. For real reflection groups these bases are cellular.
