Google AI studio, ChatGPT and Claude all support this. Google AI studio is the only one that let's you branch to a separate chat though. For ChatGPT and claude you just edit the message you want to branch from.
Support: Yes. But the UX is not optimized for this.
Imagine trying to find a specific output/input that was good in the conversation tree.
Feels like a semi-simple UX fix could make this a lot more natural. Git-style forks but for chats.