1. Fossil syncs branching and tagging info among all clones; it isn't local state as with Git. Therefore, when you're looking at a timeline, others' branches and such are named, labelled, and tagged. If you need to filter away all but your current branch, that's easy, because you have all the tags. If you need to see what others are up to in parallel with you so there isn't a big surprise at merge time, Fossil does that readily, too.

2. There's a "hide" feature in Fossil. Rather than destroy information, it simply marks a given branch as not-to-be-displayed unless you go out of your way to unhide everything. It's useful for branches that never go anywhere and thus need to be forgotten.

3. For the intermediate case, where you have a branch that shouldn't be forgotten but doesn't need to be in your face all the time, you can "close" it, meaning it still shows up in the timeline when you scroll far enough back in history to see it, but it's tagged so you can filter away all closed branches, leaving only what's active. This is the default mode for "fossil branch" without arguments, for instance, where it gives only the list of open branches. Your next command will almost certainly have a branch from that list, not something closed long ago.

I could probably come up with more, but that should get you started.