  Let the pair $(\mathfrak{g},J)$ be a nilpotent Lie algebra $\mathfrak{g}$
(NLA for short) endowed with a nilpotent complex structure $J$. In this paper,
motivated by a question in the work of Cordero, Fern\'andez, Gray and Ugarte,
we prove that $2\leq \nu(J) \leq 3$ for $(\mathfrak{g},J)$ when
$\nu(\mathfrak{g})=2$, where $\nu(\mathfrak{g})$ is the step of $\mathfrak{g}$
and $\nu(J)$ is the unique smallest integer such that
$\mathfrak{a}(J)_{\nu(J)}=\mathfrak{g}$ as in Definition 1 and 8 of the paper
by Cordero, Fern\'andez, Gray and Ugarte. When $\nu(\mathfrak{g})=3$, for
arbitrary $n \geq 3$, there exists a pair $(\mathfrak{g},J)$ such that
$\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, for which we call the $J$ in the pair
$(\mathfrak{g},J)$, satisfying $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, a
maximal nilpotent (MaxN for short) complex structure. The algebraic dimension
of a nilmanifold endowed with a left invariant MaxN complex structure is
discussed. Furthermore, a structure theorem is proved for the pair
$(\mathfrak{g},J)$, where $\nu(\mathfrak{g})=3$ and $J$ is a MaxN complex
structure.
