Does 405 nm light set off phosphorescence in the eye?
Instead of phosphorescence, a more parsimonious explanation is the perception of afterimages. Afterimages regularly appear after viewing a bright stimulus. They occur in the same location in the visual field as the original stimulus, but lack clarity. Afterimages depend on the intensity and contrast of the original stimulus (i.e., they are more pronounced with bright stimuli in a dark environment), the time of fixation (i.e., longer fixation generate more persistent afterimages), and retinal adaptation (i.e., dark environments enhance afterimages).

After images are almost always the complementary color (negative afterimage) to the original stimulus, but can very briefly be the same color (positive afterimage) when viewing an exceptionally bright stimulus. A stimulus consistently produces the same afterimage, which varies in size based on the distance between the person and the background. After images are often revived by blinking. 

Afterimages are thought to derive mainly from photobleaching of the retina, although cerebral processes may contribute too.

Phosphorescence in the retina has to be induced by injecting phosphorescent markers (Wanek et al., 2013). As far as I know, the retina is not autophosphorescent. 

References
- Gersztenkorn &amp; Lee, Survey Ophthalmol, (2015): 1-35
- Wanek et al., Curr Eye Res (2012); 37(2): 132–137