We present the result of Subaru Telescope multi-band adaptive optics observations of the complex gravitationally lensed quasar SDSS J1405+0959, which is produced by two lensing galaxies. These observations reveal dramatically enhanced morphological detail, leading to the discovery of an additional object 0. 26 from the secondary lensing galaxy, as well as three collinear clumps located in between the two lensing galaxies. The new object is likely to be the third quasar image, although the possibility that it is a galaxy cannot be entirely excluded. If confirmed via future observations, it would be the first three image lensed quasar produced by two galaxy lenses. In either case, we show based on gravitational lensing models and photometric redshift that the collinear clumps represent merging images of a portion of the quasar host galaxy, with a magnification factor of 15 - 20, depending on the model.