Afaik, swimming isn't meant to be possible in this game. That said, I've found a way to do so. Did it a couple times to make sure that I got the report correct, but...
When you flip off of (crash) your hoverboard in the right area(s), (at least in Cetus/Plains,) you will sink to the bottom of the water before your position is reset, having made contact with the bottom of the water area, it counts as having hit land, and you will be reset under the water rather than it returning you to actual land. After this, you can explore underwater, as well as use your fishing rods, weapons, items, etc.
Bait and the like, placeable items, will just kinda fall out of the map, but otherwise everything functions the same as it would on land. The camera floats above the water's surface, but will go back to normal camera offset if you go to a place that's deep enough. At some depth before the camera goes back to normal position, it will appear as normal position if you look up instead of down or straight.
