  We describe explicitly all quaternionic contact hypersurfaces
(qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion
projective space. We show that up to a quaternionic affine transformation a
qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in
$\Hnn$. Moreover, we show that an embedded qc-hypersurface in a hyper-K\"ahler
manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature
tensor of the ambient hyper-K\"ahler metric is degenerate along the
hypersurface.
