  Context. The next generation of space-borne instruments dedicated to the
direct detection of exoplanets requires unprecedented levels of wavefront
control precision. Coronagraphic wavefront sensing techniques for these
instruments must measure both the phase and amplitude of the optical
aberrations using the scientific camera as a wavefront sensor.
  Aims. In this paper, we develop an extension of coronagraphic phase diversity
to the estimation of the complex electric field, that is, the joint estimation
of phase and amplitude.
  Methods. We introduced the formalism for complex coronagraphic phase
diversity. We have demonstrated experimentally on the Tr\`es Haute Dynamique
testbed at the Observatoire de Paris that it is possible to reconstruct phase
and amplitude aberrations with a subnanometric precision using coronagraphic
phase diversity. Finally, we have performed the first comparison between the
complex wavefront estimated using coronagraphic phase diversity (which relies
on time-modulation of the speckle pattern) and the one reconstructed by the
self-coherent camera (which relies on the spatial modulation of the speckle
pattern).
  Results. We demonstrate that coronagraphic phase diversity retrieves complex
wavefront with subnanometric precision with a good agreement with the
reconstruction performed using the self-coherent camera.
  Conclusions. This result paves the way to coronagraphic phase diversity as a
coronagraphic wave-front sensor candidate for very high contrast space
missions.
