  We develop an a posteriori error estimator for the Interior Penalty
Discontinuous Galerkin approximation of the biharmonic equation with continuous
finite elements. The error bound is based on the two-energies principle and
requires the computation of an equilibrated moment tensor. The natural space
for the moment tensor consists of symmetric tensor fields with continuous
normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ)
mixed formulation. We propose a construction that is totally local. The
procedure can also be applied to the original HHJ formulation, which directly
provides an equilibrated moment tensor.
