  Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system,
$\vec{v}\in\mathbb{R}^q\setminus\{\vec{0}\}$ be a direction vector and
$\textbf{b}\in\mathbb{R}_+^{q-1}$. We study $(X,T)$ that has bounded complexity
with respect to three kinds of metrics defined along direction $\vec{v}$: the
directional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-mean
metric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric
$\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has bounded
topological complexity with respect to
$\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ (resp.
$\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$) if and only if $T$ is
$(\vec{v},\textbf{b})$-equicontinuous (resp.
$(\vec{v},\textbf{b})$-equicontinuous in the mean). Meanwhile, it turns out
that an invariant Borel probability measure $\mu$ on $X$ has bounded complexity
with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if
$T$ is $(\mu,\vec{v},\textbf{b})$-equicontinuous. Moreover, it is shown that
$\mu$ has bounded complexity with respect to
$\{\bar{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $\mu$ has
bounded complexity with respect to
$\{\hat{d}_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ if and only if $T$ is
$(\mu,\vec{v},\textbf{b})$-mean equicontinuous if and only if $T$ is
$(\mu,\vec{v},\textbf{b})$-equicontinuous in the mean if and only if $\mu$ has
$\vec{v}$-discrete spectrum.
