1. Clash of Clans stores data on its own server. If available, a single base is linked to a Game Center account.
2. You can switch between two bases by switching between Game Center accounts, and as far as I know, that requires 2 Apple IDs (What your son said)
3. There are some precautions you need to take.
4. Whenever you switch between Game Center accounts, the game prompts whether you want to delete the old base ONLY ON THE DEVICE DATA. The old base is still linked with the previous Game Center account.
5. If that prompt is declined EVER, the Game Center data of one Game Center account will be overwritten with the other account and the lost account will not be retrievable.
6. When a game is already loaded onto a device, that account stays there unless another Game Center account with a previously linked base is loaded. It does not get erased from the device if the app is deleted or you log out of Game Center.
7. To overcome the aroused problem in no. 6, you will need to separately link 2 bases to different accounts on DIFFERENT DEVICES and then keep on doing no. 1&2, while keeping precautions in mind.
Hope you did understand now.