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You wouldn't need to switch anything to display different colours.
Each pixel is divided to four subpixels which are each behind a different colour filter (R,G,B,W).
For each subpixel you can set 16 different grey levels, which after going through the colour filter are 16 different colour intensity levels.
Since the 4 subpixels are very close together, you see each such quartet as one pixel and the colours mix up to create a compound colour. 16 by the power of 3 (RGB) plus the W for intensity gives you 4096 colours in 16 levels of intensity.
The same principle is used by LCD displays.
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