I went over this topic years ago when I was teaching John's Gospel at John 3:14.

The symbol for medicine and healing is actually the Rod of Asclepius, which is a stick with a single snake (some say a parasitic worm) wrapped around it and facing outward. It appears regularly on ambulances and rescue vehicles. It predates Moses and most likely was recognized as a symbol of healing by the Children of Israel in the wilderness when Moses lifted it up. It was mentioned in an ancient Egyptian papyrus around 1500 BC.

The symbol which doctors and hospitals mistakenly use is the Caduceus, a pole with two snakes wrapped around it, heads facing the pole, and wings at the top of the pole. It was associated with the Greek god Hermes and symbolizes commerce.

Then again, maybe the doctors and hospitals know what they're doing.
rod of asclepius.jpg