  We prove an energy quantization result for Willmore surfaces with bounded
index, whether the underlying Riemann surfaces degenerates in the moduli space
or not. To do so, we translate the question on the conformal Gauss map's point
of view. In particular, we prove that in a neck or a collar region, the
conformal Gauss map converges to a light-like geodesic in the De Sitter space.
