Kecleon doesn't even do that, despite having both Protean and Colour Change. (Oh, and being a bloody Chameleon)
I don't expect Game Freak to start bothering with that now.
I was gonna say it'd be cool, but then I realised it could be harder for opponents to remember what type it is, therefore, giving you a slight advantage. (Yeah, it's a long shot, I know) So maybe it's better that it stays the same colour. After all, I have a shiny Greninja and I don't want it to change colour. Ever.