We predict the existence of long-lived and highly anisotropic plasmon modes in a two-dimensional molybdenum oxide dichloride ($\mathrm{MoO}{\mathrm{Cl}}_{2}$) monolayer. We show that along the $x$ direction (\ensuremath{\Gamma}-X), the plasmon modes are immune from the single-particle excitations as $q<0.255\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\text{--}1}$ and the maximal energy (up to \ensuremath{\sim}2.5 eV) reaches the visible region. In the $y$ direction (\ensuremath{\Gamma}-Y) the intraband plasmons decay rapidly into electron-hole pairs as $q>0.071\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\text{--}1}$, but the interband plasmons are robust in a wide range of $q$ $(0.048\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\text{--}1}<q<0.5\phantom{\rule{0.16em}{0ex}}{\AA{}}^{\text{--}1})$ with the energy varying slowly from 0.82 to 1.20 eV. We correlate these intriguing plasmons to the unique spatial symmetries of the Bloch electron wave functions near the Fermi level, which can sustain highly anisotropic plasmons and suppress the single-particle excitations. In view of the strippability of the $\mathrm{MoO}{\mathrm{Cl}}_{2}$ monolayer from the already existing bulk material, the directional plasmons in the visible region demonstrated in the $\mathrm{MoO}{\mathrm{Cl}}_{2}$ monolayer offer a tantalizing platform for nanoplasmonics.