As others have said they will on occasion dive and grab something on the bottom and drown. However, I think this is the all around last resort. I know puddle ducks and geese will always come back to shore and hide somewhere. I find a few every season just taking my dog for a walk when I'm out...he will just run up into a log jam usually and a minute or two later he has a bird.
I usually layout hunt big water. I have noticed with divers and puddle ducks that on a calm glass like day. When we cripple a bird, I'll chase him down with the boat. Almost always they will dive when I get around 70 yards or so. With that said of they don't surface I can tend to see their break just bearly break the surface. I have has success with just shooting at it, usually the birds will not be dead but dazed floating giving me enough time to scoop them up.
This is next to impossible to do Ina chop. I tend to back off the birds if I lose them and one of two things will happen. If there are seagulls in the area soon you can just watch them start dive booming something in the distance. 9 times out of 10 that is your bird they are trying to kill. Number two just backing off for a half hour or so and glassing you can usually see then resurface and begin to swim. Then the chase begins again. Lol
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The only thing worse than being blind is having sight with no vision- Helen Keller