  We study co-adapted couplings of (canonical hypoelliptic) diffu-sions on the
(subRiemannian) Heisenberg group, that we call (Heisenberg) Brow-nian motions
and are the joint laws of a planar Brownian motion with its L{\'e}vy 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 $\ge$ 0. Actually, we prove the stronger result that they can
not stay bounded in L p for p $\ge$ 2. We also study the coupling by
reflection, and show that it stays bounded in L p for 0 $\le$ p < 1. Finally,
we explain how the results generalise to the Heisenberg groups of higher
dimension
