We study co-adapted couplings of (canonical hypoelliptic) diffusions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brownian motions and are the joint laws of a planar Brownian motion with its Lévy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time t ≥ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p ≥ 2.We also prove two positive results. We first study the coupling by reflection and show that it stays bounded in L p for 0 ≤ p < 1. Secondly, we construct an explicit static (and in particular non co-adapted) coupling between the laws of two Brownian motions, which provides L 1 -Wasserstein control uniformly in time.Finally, we explain how the results generalise to the Heisenberg groups of higher dimension.