I saw a couple other posts about this so I thought I would post my ideas in hopes the developers will see it.
It would be awesome if your companion would fish with you. The easiest way I imagine would be for them to simply use the same animation that NPCs have that fish and when you are standing still fishing have the companion walk to the water and pretend to fish with you.
However it would be even more amazing if they also gave you the bonus chance that fishing with another player did. I think letting them fish and getting their fish would be too far, but if they just gave us a bonus of another person fishing the same spot that would be useful without being broken.
I made this post because I have spent hours fishing with my companion just standing there, even a simple fishing emote from your companion would help so much with immersion.
PS having them sit when you do would also be awesome.