  Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for
$\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space
$X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) /
\mathrm{PU}_n$. If $\Omega$ is a compact subset of $X$, then we prove the bound
$\|\phi|_{\Omega}\|_{\infty}\ll_{\Omega} \lambda_{\phi}^{n(n-1)/4-\delta}$ for
some $\delta>0$ depending only on $n$, where $\lambda_{\phi}$ is the Laplace
eigenvalue of $\phi$.
