Plumes generated from a point buoyant source are relevant to hydrothermal vents in lakes and oceans on and beyond Earth. They play a crucial role in determining heat and material transport and thereby local biospheres. In this study, we investigate the development of rotating point plumes in an unstratified environment using both theory and numerical simulations. We find that in a sufficiently large domain, point plumes cease to rise beyond a penetration height $h_{{f}}$, at which buoyancy flux from the heat source is leaked laterally to the ambient fluid. The height $h_{{f}}$ is found to scale with the rotational length scale $h_{ \!{ f}}\sim L_{ \!\textit{ rot}}^p\equiv ({F_0}/{f^3})^{{1}/{4}},$ where $F_0$ is the source buoyancy flux, and $f=2\varOmega$ is the Coriolis parameter ($\varOmega$ is the rotation rate). In a limited domain, the plume may reach the top boundary or merge with neighbouring plumes. Whether rotational effects dominate depends on how $L_{\textit{rot}}^{p}$ compares to the height of the domain $H$ and the distance between the plumes $L$. Four parameter regimes can therefore be identified, and are explored here through numerical simulation. Our study advances the understanding of hydrothermal plumes and heat/material transport, with applications ranging from subsurface lakes to oceans in icy worlds such as Snowball Earth, Europa and Enceladus.