But you can't have more than one companion.
Wrong.
So I was taking that stupid dog, Barbas, to get that stupid axe and a stupid dragon attacks. My awesome mage companion throws a fireball. I assume some splash damage hits the dog and after they're done with the dragon they start attacking each other.
How can I get them to stop and be friends again?
You can use a Calm spell on them to stop them fighting. If you don't have Calm (or a staff or scroll or weapon enchanted with calm) you're going to have to fast travel (if you can) to somewhere else, or enter a building etc. Anything to make you go through a loading screen. When you come out on the other side, your two companions should be pals again.
(They shouldn't be able to kill each other, so maybe letting them fight it out will cause them to reset when one of them kneels down and yields.)
If you are on the PC version, which I'm assuming with multiple companions that you are, you can disable/enable the "angry" companion.
To do this:
1. open the console (~),
2. click on the companion, you'll get a code listed at the top center of the console,
3. type "disable" and hit enter,
4. exit console (~),
5. wait for the other companion to relax,
6. reopen console,
7. type "enable" and hit enter.
Note: Be sure that you've clicked on the companion and not some other element in the world. If your companion didn't disappear when you typed disable, immediately type enable and then retry to click on the companion, you may have to get closer.
This is good for if you don't have a calm spell or a means to cycle a loading screen.
I slept in a bed roll to stop mine from fighting. It may also work with the wait function.
