I have often wondered about seeing in different wavebands - I suspect that's why I've equipped myself with gear that can convert different frequencies into something that I can perceive with my eyes. Would I want to swap my present eyes for a different band? No, but if I could augment my present senses with additional bands I'd have a good think about it.
I have looked at the world in LWIR, MWIR, NIR, UV and X-Ray I can fairly confidently say that most of the 'useful' information is packed into that small sliver of spectrum we refer to as 'visible light'. Next up is probably near-UV, and there are many species of insect, bird and (weirdly) fish that can perceive UV, some as a '4th colour' and others in conjunction with a smaller selection of lower visible bands.
Once you move to near-IR (from 'just below dark red' to about 1.1µm), you find there aren't really any 'colours' in the way we perceive them. Sure, some things are brighter or darker in those bands - the sky is black and foliage is bright - but I haven't generally noticed much difference between, say, 1100-900µm and 900-750µm. There will be odd instances of course but nothing like a peacock's feathers, a butterfly's wings or a garden full of flowers. I am not aware of any organism that has evolved particular sensitivity to those bands.
MWIR is a wide band and does have some interest, as there are a number of things that have sharp absorbencies within that range. It might be interesting to be able to see in that band but it'd have to be split into multiple separate 'colours' to be useful. Again, nothing I'm aware of has any senses in that band.
LWIR, again, is only slightly interesting. Yes, you can tell the temperature of household objects and could, probably, map different sub-bands to (say) R, G and B - but mostly all you'd see is that hotter things were bluer. Would this be more useful than our thermal cameras that just show hotter things as brighter? I don't know. It is interesting to know that a few organisms can - to a limited extent - 'see' in LWIR. Some snakes have specialised thermal receptors that let them detect prey. And, of course, you'll be well aware that your own skin has some thermal reception capability - your face can easily detect from some inches the thermal radiation from, say, a really fresh cup of tea. Hotter things like bonfires can be sensed much further away.
I haven't thought much about looking at lower frequencies than LWIR, though there might be some interesting stuff there if today's relatively primitive experiments with terahertz radiation are anything to go by. Below that and you're into radio of various sorts. I wonder what an electromagnetically-sensitive alien would make of our planet's incredible level of noise in the RF spectrum.
Back up now to UV and beyond. Yes, flowers look different in UV, as many include patterns that have co-evolved with pollinators' eyes to guide those insects to the 'good bits'. A very few humans can see by near-UV; there is a rare genetic condition that adds sensitivity to a fourth colour and an also-rare eye condition where the lens is missing, thus allowing UV to reach the retina (the lens is fairly opaque to UV and thus helps protect our retinas from damage).
Moving up the spectrum brings us into 'vacuum-UV', so-called because it won't propagate in air due to heavy absorption and would thus probably be useless as a visual sense.
Then you start getting into soft X-Rays. These would let you see through some less-dense things. As the energies increase to, say, 50keV you reach the area where you look right through skin and see skeletons - that's where medical imaging lives. But there aren't that many natural X-Ray emitters on our planet so even if you could see that range it would be pretty dark. But lift your (very sensitive) X-Ray eyes to the heavens and it would be a different story: there are numerous huge X-Ray emitters across the vastness of space so the night (and day?) sky would look very different. (As an aside, there are also things to see in the universe at other wavelengths from radio upwards; I just conveniently skipped over them in the earlier parts of this muse).
X-Raye give way to alpha, beta and gamma radiation, all of which have increasing penetrative power. If you wanted to be able to look straight through (say) a large ship or deep into the ground then gamma-ray vision would be quite interesting I suspect. But for everyday use you'd simply see straight through day-to-day objects - and anyway, many would be damaged by the sorts of levels of gamma radiation necessary to form any kind of useful image.
So there we have it: as far as I know there's quite a lot of 'nothing to be seen here' across the spectrum. Most life has evolved to use the same tiny sliver of 'visible light'. Even when you look at things like deep-water lantern fish, which inhabit depths where daylight can't penetrate and create their own light to lure prey - they light up in the same spectrum we see in. I can't imagine there's any reason to have prevented their biology from evolving to produce, say, IR or UV light (and the prey to have similar sensitivity), but that's just not how it is.
I think my final conclusion is that if we had a set of terminals on the side of our heads that let us bolt on supplementary wavelength sensors I'd probably go for a wide-range LWIR-to-near IR option - but as I can't envisage anything like that happening soon I'll happily stick to my toys and the Mk 1 eyeball.