The cliche is that Bowie is a chameleon; the myth is that chameleons change color to fit in with their surroundings.
In The Book of General Ignorance, John Lloyd & John Mitchinson write,
"Total lie. They change color as a result of different emotional states. If they happen to match the background it's entirely coincidental. Chameleons change color when frightened or picked up or when they beat another chameleon in a fight. They change color when a member of the opposite sex steps into view and they sometimes change color due to fluctuations in either light or temperature."
In this way, David Bowie is totally like a chameleon: shifting in response to an alarming environment.
In an interview in 2003, Bowie said,
"My entire career, I've only really worked with the same subject matter. The trousers may change, but the actual words and subjects I've always chosen to write with are things to do with isolation, abandonment, fear and anxiety---all of the high points of one's life."
Perhaps the only sane response to excessive stimulation & rampant alienation is constant change. Splintered by schizophrenic reality, you hurtle through a flurry of hues, shapes, textures, belief systems, philosophies, movements, hair styles, costumes, literary modes, sexual orientations, technological innovations, on the search for peace amongst the pieces, only to discover that your skin was yours all along, channeling the flux inwardly, constructively, to combat the anxiety of external disorder.
The placement & coloration of his eyes, too, evoke the chameleon.
More informative chameleonic fun from General Ignorance:
"A chameleon's skin contains several layers of specialized cells called chromataphores---from Greek chroma (color) and pherein (to carry)---each with different color pigments. Altering the balance between these layers causes the skin to reflect different kinds of light, making chameleons a kind of walking color wheel...The word chameleon is Greek for 'ground-lion.'"