I've never been to the Punchbowl, but I have done research on it. It is a very rugged depression in the earth several acres in size and is deffinitely NOT a place to go alone. It's located beside the Mississippi River and was used as a hiding place by river pirates, Trace bandits and other no-goods. I believe the Indians stayed away because they were superstitious. The old time river boat pilots reported that their compasses would deflect everytime they pasted that place.
I live about 50 miles away from Natchez and it's been several years since I've been in the area.
When you get the chance, go to the town library and look for books about Natchez in the earlier years. Don't look for books with the word "treasure" in the titles. Get into the history section and read the older books published in the 1950s and OLDER. I found an interesting one written by a black freedman who was a barber in town. I believe the title is The Barber of Natchez. I would think that the Natchez library would have a fantastic local history section.
Be aware that Natchez is the oldest, continuously inhabited place on the Mississippi River system, which means that the possibilities are almost endless for gold coins, jewelry, etc. One last thing. The Under-the-Hill area has been washed away by the Mississippi River so many times through the years, it would probably be a waste of time to do any searching there.