We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra
g
\mathfrak {g}
under the presence of a complex structure
J
J
. In particular, we find a bound for the dimension of the center of
g
\mathfrak {g}
when it does not contain any non-trivial
J
J
-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras
g
\mathfrak {g}
admitting this particular type of complex structure
J
J
. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs
(
g
,
J
)
(\mathfrak {g}, J)
.