There are a number of odd bits in the image - the 'jewelry' + watch on one of the hands, the strange hair effect, the overlap of what appears to be a female's blouse... It looks like there may be a woman seated behind him, and that perhaps it is a double exposure from 2 flashes hence the overlaps of some items that moved between the flashes. However, the image is way too small to examine properly, hence my request for a bit more context right at the start - other images that were taken at the time may reveal the other person or give clues as to what has happened, or if Buzz really does have >2 hands and a proclivity for strange jewelry and hairstyles..
Now that you mention it, there is a fourth hand tossing his long main of silver hair. It no wonder he was picked for the first moon landing. Being able to throw all those switches and breakers so fast must have been a big help. Maybe he is really an avatar of Vishnu?
For those who wish to maintain an illusion, ignorance is the best source of knowledge.
Enlarging the photo and lightening it a little makes things clearer.
Buzz's wife, Lois, is shorter than Buzz and has white hair, so it's possibly her sitting on his right and perhaps adjusting her hair with both hands. Her right hand doesn't seem to be holding anything and just coincidentally looks as if it might be holding a cellphone to Buzz's mouth. It is further from the camera than Buzz's face as it is a little more out of focus.
The main indicators that she's adjusting her hair are her left ear, immediately behind Buzz's collar, and her left hand behind and to her left of the ear and looking a little like it may be brushing or holding hair.
There appears to be only one flash exposure from a flash unit which is almost directly above the camera lens. It has cast a shadow of the nose of the man on the left down and slightly to the left, and Buzz's hair casts a small, faint shadow down and to the right onto Lois's hair and ear.
Don't criticize what you can't understand. — Bob Dylan, “The Times They Are A-Changin'” (1963) Some people think they are thinking when they are really rearranging their prejudices and superstitions. — Edward R. Murrow (1908–65)