Here's one problem with any camera that attempts to capture such a broad range of light: The display medium must be able to show what it has captured in only the visible spectrum because if it displayed the UV in the UV spectrum and/or the IR in the IR spectrum you wouldn't be able to see it. This means the colors in the visible spectrum would need to be shifted to make room on either end for the UV and IR.
The main problem, though, is the way that most digital cameras detect color. They use a Bayer mask that is most sensitive to very near the same wavelengths of light that the cones in our retinas are. Even if IR and UV weren't filtered prior to the light reaching the sensor, the sensitivity curve of the sensors would mean the visible light would dominate the image to the point the UV and IR would have very little influence (other than the IR might cause highlights to blow at slightly lower exposure levels).
Any light past the red end of the visible spectrum invokes a fairly equal response from sensels under all three filtered colors in most digital cameras. The sensors used in typical digital cameras are also unable to discriminate very well between various wavelengths on the UV end. Because most digital cameras are designed to reproduced visible light only, the sensors have filters in front of them to block infrared and UV light from reaching the sensor.