@Mayank - our pump runs all winter long, waterfall and all. There are factors that make this possible - for example, our pump is six feet below ground, underwater. Our plumbing is all buried - nothing is subject to freezing. And our waterfall is wide and concave, so the ice that forms won't re-direct water out of the pond. We don't have a skimmer box or a bio filter that could crack. So the advice you were given might indeed apply to your pond. Ours has run this way for 8 winters with no trouble.
Well, no trouble EXCEPT last year... we have had issues with our pump plumbing. The pump is in a vault and does not sit on the bottom, so it was creating some downward pressure on the plumbing line where it connects the pump to the main pipe that goes to the bog. It has separated on us twice - once was mid-summer, so no big deal. We just pulled it out and glued it and it was done. We assumed it was just a bad glue job - in fact the old plumber who works in the plumbing department at our local hardware store told us it HAD to be a bad glue job as he had never seen a PVC glue weld fail. So we went with that assumption.
The second time it happened was right before the new year - dead of winter. It didn't let loose completely, but it was leaking around the connection and wasn't pushing the full volume of water. We would have left it alone except we had a string of really nice warm days in early January that year so we decided why not? Well I'll tell you why not - because even when the air is warm, the plumbing and the water are still cold. We disassembled the plumbing, glued it back together and... could not get it hooked up again. We later realized the frost in the ground pushes the main line up just enough that it made the connection impossible.
So due to that completely fluky happenstance, the pump was off from January 10 or so until early March. We ran our aerator, but it just wasn't enough for our big fish. As the pond started to thaw, we could see they were all dead. It was indeed very sad and was avoidable had we only not made the wrong assumption about the plumbing failure the first time around.
When it was warm enough, we hired a professional to come and hook it up and make a few tweaks so it won't happen again.
So that's my sad story. And my recommendation is if you have too many big koi in a pond, your best bet is to keep the water moving. Better recommendation - keep your pond understocked. Or keep goldfish. We had three dozen goldfish when this disaster happened... they were unscathed.