First, I like your question. You might ponder what you imply when you want to "see" the common parts of the electromagnetic spectrum (e.g. visible light, radio waves, x-rays etc.)
If you want a camera that can "see" a broader spectrum, you are asking for a camera that can transform a broad spectrum of wavelengths into visible light (since we can only use visible light to interpret what the camera is picking up).
Think of this logic: I create a camera that is sensative to visible and infrared radiation. The camera transforms this combined spectrum range into visible light. Problem: if I am seeing the full visible light spectrum, where do I have any unique wavelengths of visible light left to represent infrared wavelengths? Therefore, you have some limitation on overlapping spectrum ranges with visible light. Maybe you'd see greater intensity or brightness in some colors.
Of course, there are other ways to analyze electromagnetic radiation, like in terms of a spectrogram, we can tell which frequencies are more "present" in a signal. But analyzing electramag radiation in this manner for a large spectrum has some harsh limitations: 1) you'd need a lot of different antennas (one size does not fit all) 2) an insanely fast sampling frequency for higher frequency radiation 3) a large bottle of aspirin for the headache you'd get trying to process all that information.
I suppose it would be possible for a hypothetical biological eyeball to have sensitivity to a larger band of the spectrum. You'd need a brain that could process the information. It is possible, since some creatures like Butterflies can make good use of ultraviolet radiation.
Unfortunately I think most engineers on this website have less experience designing eyeballs (excluding some ninja bio-engineers if they exist).