From the little research I have found, mostly on other species - it is probably safe to extrapolate that they can see moment extremely well, but not color - but they seem to see short wave lengths (red and infrared) better or at least with better clarity than long wave lengths (blue and UV). Some of the head-bobbing may be to focus direction and shapes, birds of prey do the same thing before striking. This is an excerpt of one of the papers that I read discussing pupil function for the paper "The pupils and optical systems of gecko eyes" by
Lina S. V. Roth; Linda Lundström; Almut Kelber; Ronald H. H. Kröger; Peter Unsbo.
[Eyes adapted for vision at night, such as the eyes of nocturnal geckos, with a large pupil and a short posterior nodal distance (here also called focal length, f), are especially affected by longitudinal chromatic aberration. As a result, light of short wavelengths is refracted more strongly and thus focused closer to the lens than light of long wavelengths. If this is not corrected for in an eye adapted for nocturnal vision, the retinal image is severely blurred. Multifocal optical systems with distinct concentric zones of different refractive powers have been suggested to correct for some of the defocus on the retina caused by chromatic aberration (Kröger, Campbell, Fernald, & Wagner, 1999). Kröger et al. have shown that the eyes of the nocturnal gecko, Homopholis wahlbergi, have multifocal optical system.
In addition, the light-adapted pupils in nocturnal geckos are different variations of vertical slit pupils. Apart from the effectiveness in shutting out light during the day, the mode of constriction of slit pupils has been suggested to be of advantage in multifocal eyes, since it allows for all refractive zones of the optical system to be functional at all states of pupil constriction (Kröger et al., 1999; Malmström & Kröger, 2006). We investigated the pupil dynamics and the multifocal optical system of helmet geckos to see whether the light-adapted pupil allows for all concentric zones of the optical system to refract incoming light.]