A complex lattice consists of a pair of $\mathbb{R}$-linearly independent vectors of the real-vector space $\mathbb{C}$. We call two lattices $(\lambda_1,\lambda_2)$ and $(\mu_1,\mu_2)$ equivalent if $\mathbb{Z}\lambda_1+\mathbb{Z}\lambda_2= \mathbb{Z}\mu_1+\mathbb{Z}\mu_2$. Is there a classification of equivalence classes of complex lattices? In particular I am interested whether any lattice admits an equivalent lattice of a "nice"(to be specified) type? That is, whether there are "canonical" representatives of any equivalence class of lattices? Certainly for any lattice there is an equivalent lattice $(\mu_1,\mu_2)$ with $Im(\mu_1)\geq0$ and $Im(\mu_2)\geq0$ (by reflecting at $0$, if necessary). But can the position of $\mu_1$ and $\mu_2$ be even further restricted?
Online I found mention of the crystallographic restriction theorem, which might be relevant. I have not been able to find a good source on this theorem though. So I'd also be happy about a good reference.