The magic of fisheye
Do we actually know if fish see the world like this? I am not sure. But fisheye photography has something magic, if used correctly. Fisheye photography does not have to be circular. A rectangular 3:2 aspect ratio, filled, can still give you a strong fisheye impression.
Depending how you position your object, the centre can move close to you or seemingly away.
I am aware that it is possible to create fisheye effects in software. Generally they look ok, but it is not the same as with a dedicated lens. The subtle distorions, the out-of focus areas, the light reflections, they are nether there with software rendering.
Here are some images, mostly shot with a Nikon 10.5mm lens on either a Nikon D7000 or D750.