  We study the algebraic constraints on the structure of nilpotent Lie algebra
$\mathbb{g}$, which arise because of the presence of an integrable complex
structure $J$. Particular attention is paid to non-abelian complex structures.
Constructed various examples of positive graded Lie algebras with complex
structures, in particular, we construct an infinite family $\mathfrak{D}(n)$ of
such algebras that we have for their nil-index $s(\mathfrak{D}(n))$: $$
s(\mathfrak{D}(n))=[ \frac{2}{3}\dim{\mathfrak{D}(n)} ]. $$
