  Quantum vortices in superfluids have been an important research area for many
decades. Naturally, research on this topic has focused on two and
three-dimensional 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-Pitaesvkii equation, that the
simplest case of equal-frequency double rotation can stabilise a pair of vortex
planes intersecting at a point. This opens up a wide number of future research
topics, including unequal-frequency double rotations; the stability and
reconnection dynamics of intersecting vortex surfaces; and the possibility of
closed vortex surfaces.
