[Buteo]

I'm sorry, Monica. I really wasn't trying to shoot down yer clip.
It's just I've seen that clip over a year ago & it was discussed on a bunch of the Aquarium forums I was on. The concensus was that it was faked.

The first of those other linkd is fully believeable.
Our one Oscar, Blondie loved being petted my me. I'd stick my hand in the tank & she would swim over, rubbing up against my palm & between my fingers.

But as pertains to that goldfish clip...
It's not that fish can't be trained.
Even Adam & Jamie, the Discovery Channel's "Mythbusters" found you could train goldfish to work through a maze to reach food.
But let's be honest.
There's a big difference between getting a single fish to do a certain "trick" and getting groups of fish to swim perfectly syncronized.
Even fish that normally do school up don't swim together that well, especially in such a small group.

Watch the clip again. For the most of it, the fish maintain the same distance apart & use the same fins.
Speaking about fin motion: I've seen many, many goldfish. They're "tail swimmers", rarely using their pectoral fins as a main means of propulsion. The fish on that clip on the other hand hardly ever use their tails. If you watch really close, you can see the exact same moves made by one fish are duplicated perfectly by one of the others...right up to making duplicate ripples on the surface.
Knowing a thing or 2 about video SFX, it would be much easier to fake that clip than it would be to train the fish to do that.

What these things do show is that saying about "goldfish only having a 10 second memory" is hogwash.