We present spectroscopy and laser guide star adaptive optics (LGSAO) images of the doubly imaged lensed quasar SDSS J1206+4332. We revise the deflector redshift proposed previously to z d = 0.745, and measure for the first time its velocity dispersion σ = (290 ± 30) km s−1. The LGSAO data show the lensed quasar host galaxy stretching over the astroid caustic thus forming an extra pair of merging images, which was previously thought to be an unrelated galaxy in seeing limited data. Owing to the peculiar geometry, the lens acts as a natural coronagraph on the broad-line region of the quasar so that only narrow $\left.\mathrm{C {\scriptscriptstyle III}}\right$ emission is found in the fold arc. We use the data to reconstruct the source structure and deflector potential, including nearby perturbers. We reconstruct the point-spread function (PSF) from the quasar images themselves, since no additional point source is present in the field of view. From gravitational lensing and stellar dynamics, we find the slope of the total mass density profile to be γ′ = −log ρ/log r = 1.93 ± 0.09. We discuss the potential of SDSS J1206+4332 for measuring a time-delay distance (and thus H 0 and other cosmological parameters), or as a standard ruler, in combination with the time-delay published by the COSMOGRAIL collaboration. We conclude that this system is very promising for cosmography. However, in order to achieve competitive precision and accuracy, an independent characterization of the PSF is needed. Spatially resolved kinematics of the deflector would reduce the uncertainties further. Both are within the reach of current observational facilities.