Quantum vortices in superfluids have been an important research area for many decades. Naturally, research on this topic has focused on two-dimensional (2D) and 3D superfluids, in which vortex cores form points and lines, respectively. Very recently, however, there has been growing interest in the quantum simulation of systems with four spatial dimensions; this raises the question of how vortices would behave in a higher-dimensional superfluid. In this paper, we begin to establish the phenomenology of vortices in 4D superfluids under rotation, where the vortex core can form a plane. In 4D, the most generic type of rotation is a double rotation with two angles (or frequencies). We show, by solving the Gross-Pitaevskii equation, that the simplest case of equal-frequency double rotation can stabilize a pair of vortex planes intersecting at a point. This opens up a wide number of future research topics, including into realistic experimental models; unequal-frequency double rotations; the stability and potential reconnection dynamics of intersecting vortex surfaces; and the possibility of closed vortex surfaces.