Do anyone else's fish do these?
1. My husband and I have noticed that when we come up to the tank to just watch the fish, they actually stop what they're doing, calmly swim over to the side of the tank, and watch us.  Sometimes they beg for food, but mostly they just *stare*.  It's kind of cool and creepy at the same time.
2. When we watch the fish, we've been in the habit of talking to them (silly stuff like you'd talk to your pets).  Now one of them, and sometimes the second-smartest one, will start moving their mouths along with what we're saying!  It's not a food-searching thing or just normal movements- he actually speeds up his usual open-close to match ours.
3. I think we taught the "talking fish" to wave.  When we go up to the tank, we usually wave to the fish.  Now that one is fluttering his side fin at us- again, this is not normal swimming movements, and we haven't noticed it till this week.
The crazy thing is that they only do this when they see us watching them.  Are the fish mimicing us, or have we lost our minds and we are just imagining that our fish (two shubunkins and a comet) are smarter than they are?  They're perfectly healthy, good water quality, etc.

awww.. i wish my fish were that smart...

my two goldfish are extremely smart as well. My one fish, Timmi(we are not sure which species she is, no one can figure it out!) absolutely loves us, especially my dad. He will put his hand in the tank and she will swim right into it so he can rub her on her chin. She loves it! lol. It's the most adorable thing ever. I have sat and talked to her as well and she will mock what i do with my mouth as well. She loves to put on a show for us as well...doing flips and trying to swim really fast around the tank (she is a BIG round-bodied fish, so her swimming abilities are a little rough! lol). My male oranda Norm is also very smart. But he's more of a keep to himself kinda guy. He has a routine he does to get us to give him food. The poor guys wen is so overgrown though he can barely see us anymore! We call him Mr. Magoo, because he looks like him. Haha. So i don't care what people say, Goldfish are amazing fish and they do have personalities and intelligence. 

Just came across this post.

Yes, I believe Goldfish have intelligence.
Humphrey and Hepsiba do the same: whenever one of us is near the tank they stop what they were doing ( usually re arranging the gravel layer  ) and come to the glass as if they want to talk.
Today we have Country and Western Music going ( luv it! ) and did a bit line dancing, guess what?  our fishies moved and wiggled quicker the more we moved.

I also believe that they do have a watch on their fins, lol. When we are supposed to leave for work about 7 am they swim towards the corner of the tank which points to the door ( sure they recognize if I am dressed in work uniform ) and they are there in the evening when we come back home. I need to drop my handbag on one of them bar stools, they come to the front surface corner, half their head out of the water, opening their little mouth waiting for food to come