Ahh, understood. Today you can install the iPad version to your mac, not the best experience but would get you what you want. I'll do some more thinking about a more Mac OS native experience.
Oooo nice, not really yet... Claude stores your session history in ~/.claude/projects/-Path-to-git-repo on your local machine. I should definitely add a "teleport" command like the one claude itself has which let's you move sessions between machines. It's a little tricky with git state & un commited changes but definitely possible. I'll consider this an upvote for that feature.
(1) This one is on the top of my list. I have a branch that sets up proper iOS notifications. Coming soon.
(2) Yeah, I've considered making an actual desktop version. A hacky one exists already. You can run `catnip run` from your desktop and it will run catnip in a Docker container on your machine. There's a web interface, and it even works with native MacOS notifications.