Let $X/ \mathbb{C}$ be a K3 surface with complex multiplication by the ring
of integers of a CM number field $E$. Under some natural conditions on the
discriminant of the quadratic form $T(X)$, we produce a model $X^{\text{can}}$
of $X$ over an explicit abelian extension $K/E$ with the property that
$\rho(X^{\text{can}}/K) = \rho(X / \mathbb{C})$. We prove that $X^{\text{can}}
/ K$ is canonical in the following sense: if $Y/L$ is another model of $X$ such
that $\rho(Y/L) = \rho(X / \mathbb{C})$, then $K \subset L$ and
$X^{\text{can}}_L \cong Y$. If $E$ is fixed, our theorem applies to all but
finitely many surfaces with complex multiplication by $E$. In case $X$ is not
one of those, we still provide necessary and sufficient conditions for a model
enjoying the same properties of $X^\text{can}$ to exist. As an application to
our work, we give necessary and sufficient conditions for a singular K3
surfaces with CM by the ring of integers of an imaginary quadratic field $E$ to
have a model with all Picard group defined over $E$, and provide an alternative
proof of a finiteness result obtained by Shafarevich and later generalised by
Orr and Skorobogatov.