While I do not agree with the captain's call for the reasons above I wold probably jump in after him to be his rescue buddy in case he needs it.
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Sorry to be a "wet blanket", but: It is very difficult to estimate distances over water (my previous career was as a ship's navigator). At 1/2 mile, unless the water is totally flat, and the diver has a safety sausage, and the diveboat has a fairly high deck.... it is nearly impossible to see a diver in the water at 1/2 mile range, without binoculars and a good deal of luck. Also, in general, you wouldn't be able to hear him yelling at that range... a very good whistle or an air horn, plus the wind in the right direction and engine turned off, but probably not just a voice....
Anyway, distance estimation aside, unless you are certain you can reach the diver faster than the boat can, it would have been better to wait for the anchor to be raised, slipped or cut. And if the decision was made to swim, the captain should have stayed on the boat! He is the one most qualified to operate it, and if he were injured or needed to be rescued, it could be a big problem.
Best wishes.
Leadturn, you are dead on. If you could hear him, you were not 1/2 a mile away.
But lets assume it is 1/4 mile...200 yards... makes very little difference. First, because it is a down current swim and second because you are not doing a rescue...someone goes over there to calm the "excited person" down. Been thru that exersize a bunch of times.
You swim over, calm the person down and wait for the boat to get there. If they were really drowning, you would not get there in time.. well, unless they have invented a new form of slow, very slow drowning.
So people just don't do well by themselves, alone in the ocean.
A situation in which a diver is caught in a current and drifting away from the boat is always a difficult problem. Each situation, however, must be evaluated on its own.
Some things to consider:
A diver in a one knot current (about as fast as a person can swim with MF&S without scuba for any distance) will carry a person on the surface 1/4 mile in 15 minutes.
It may take several minutes to get the boat ready to move to chase the diver down. Grannie and trail lines must be taken in; any deco bars, hanging regulator lines and the like must be pulled up; anchor lines or moorings must be cast off. Few captains would "cut" an anchor line; they would untie it and attach a fender or other float to the end for later recovery.
It is a very dangerous proposition for the captain to enter the water to go after someone. I am aware of at least one instance here in Lake Michigan where such action caused the death of the captain - The diver survived.
A diver who is yelling is not really in immanent danger. Like a baby who just fell down and is crying, they are upset and demanding attention but not seriously injured. Someone who is drowning does not make noise.