I have the Sigma 4.5 for crop-sensor cameras. It's a bludy expensive bit of novelty 'indulgence' TBH, but it's the only full-round for APS-C bodies, Choice is better for full-frame, You have the Manual Focus Samyang or the AF Sigma, that both deliver the full round, both axis 180 Deg FOV, but it's still not exactly a huge selection to choose from!
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My personal irrationality for buying was that I picked up a 'cheap' 12mm fish for my film cameras way back when; and caught the fish.. but, always felt short changed by it not delivering the full round and a mere 165Deg Fov in top & bottom cropped circle... so in for a penny, in for a squid, may as well go all the way...
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Thing is that a full-fish is a VERY demanding lens to get results with. I can see why fish are dismissed by so many as a mere novelty, the occasions that they can 'work' is restricted, and you really do have to work at it to get it. For 'wide' situational shots about the paddock at a Kart meet? Well, I suspect opportunities wont be much different to those I get at bike events.. where I can happily walk around with a UWA on the front, and the Fish never comes out the bag.. and if it does? Display shot will usually end up a lot less 'fishy' by the time I've straightened & cropped it to make something that has the desired impact when viewed..
So, to detail the niggles of a full fish.
First is that the full fish puts a circular image into your frame. 1/3 of your frame area is masked, so you are loosing 1/3 of your pixels before you start; and as mentioned, its likely that to make a display picture with intended impact from your capture, you are likely going to crop out of even that. Square out the circle, will have about half the pixel count of the full frame, which does start to impinge on post-process potential.
Next up; the fishe eye shrinks an enormous amount of real-estate into that 1/3 masked view finder; so the lens makes stuff very small very fast, AND it's making it even smaller in the restricted viewfinder. This makes composition rather more tricky than needs be... AND it's awkward enough to start with, due to very small changes in camera angle / position having such huge effect on the perspective & distortion of the picture.
Then we have the matter of focusing; With such a short focal length, the near focus distance is often only a couple of inches, and DoF enormous. Even at just f4 you can almost have front element to infinity DoF! In fact the 12mm fish for my film cameras was actually 'fixed focus', factory set to hyper-focal distance, relying on the huge DoF to keep everything sharp! I have the AF switched 'off' on my Sigma, and use it pretty much as focus free, set to hyper-focal distance, and accept the small loss in close focus at wider apertures. It saves the horrible focus hunting! With such a wide FoV crammed into the viewfinder, the focus matrix is looking all over the shop for something to get a bead on.. and there's a lot there, and its all small, and often moving; so it does 'hunt', which gets annoying. This does beg the question why, with a fish you would want even Manual Focus, let alone Auto-Focus, or to pay the extra for it
Follow on from that is metering; much like AF and for same reason, with 1/3 the frame blacked out, then so much crammed into the FoV, and likely an enormous contrast range, where its almost impossible to exclude hot spots by careful framing, that hard to see it all cramped view-finder.... metering can similarly get a little over wrought! DO NOT rely on TTL metering with a fish! An incident reading or just a guesstimate by f16-Sunny, is often far more reliable!
With all the niggles and difficulties of working with a full fish most exacerbated by one, it would be diving into the deep end of the fish-pond to go for one straight off; and likely to put you off as encourage you. I love my 12mm fish on film; it 'only' has about a 160Deg FoV and cops the top & bottom off the circle, but you get enough 'fish effect' from it, and with less masking and less FoV you don't get quite the same extreme of niggles and it is 'easier' to work with.. and as said, most captures will be worked in post to get the impact from the display image, so likely be cropped as much anyway. I think it was a good place to start and get a feel for the things.
Begs suggestion that the relatively 'cheap' Manual Focus Rokinon 12mm, is probably as good a place as any to start. I did come across a Russian 12mm Fish, when I was shopping, that I recall was very cheap, but cant recall it's name, but could be the cheapest toe in the water start point, for likely limited use. Then there's the Manual Focus Samyang 12, which is widely regarded as good VFM. Or there's Nikon's 10.5mm Fish, which, performance wise aught to be beyond question, that offers a 'bit more fish' without going all the way; but is an awful lot of money for such a specialist bit of kit. And then you are down to the Full Round offerings, and the Samyang for VFM, sacrificing auto-focus to the Sigma.
Bottom line really is how much are you prepared to spend, and just how much fish do you think you can handle. It really is an 'occasional use only' lens, they are all much more demanding to get results with than an UWA, and the full fish is very very demanding in deed. Can you justify the spend for likely use / results?
But I'd urge cation and conservatism, and suggest you trawl the small adds for 2nd hand, and see whats on offer, rather than target a specific lens, when there's so few to really choose from anyway, and suggest you er towards a 12mm crop-fish rather than a full-round, as a start point, and see how you get on with it and how much you use it, to direct whether its worth spending more money on anything 'more' fishy or technically 'better'.