Let $K$ be a number field with ring of integers $O_K$.
If $K$ is totally real, then $O_K$ is a lattice in $\mathbb R$.
If $K$ is imaginary quadratic, then $O_K$ is a lattice in $\mathbb C$.
If $K$ is of degree $3$, then I want to consider $O_K$ as a lattice in $\mathbb C$. Is that possible?
What do you mean by "$O_K$ is a lattice in $\mathbb R$" when $K$ is totally real.
E.g. if $K = \mathbb Q(\sqrt{2})$, then $O_K =\mathbb Z[\sqrt{2}]$ is abstractly isomorphic to $\mathbb Z^2$ as an abelian group, but it is dense as a subset of $\mathbb R$ (so not one would normally call a lattice).
If the degree of $K$ is $d$, and $K$ has $r_1$ real embeddings and $2 r_2$ complex embeddings (so that $r_1 + 2 r_2 = d$), then $O_K$ may be regarded as a lattice (i.e. a discrete copy of $\mathbb Z^d$) in $\mathbb R^{r_1} \times \mathbb C^{r_2}$.