[FEATURE] Add a persistent, at-a-glance machine indicator for multi-machine Claude Code sessions

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

I run Claude Code sessions across several physical machines at the same time (a Mac and multiple Windows PCs on my local network), each doing different, unrelated work. Right now, nothing in the session list or the open tab reliably tells me at a glance which physical machine a given session is actually running commands on — the only signal is whatever text happens to be in the session title, which is easy to miss, forget to set, or lose track of when several tabs look visually similar.

This is not just a minor annoyance — it's a real safety issue. I had downloaded two large AI models on a Windows PC, then had a separate Mac session open (with plenty of memory to spare — the model would have run there without any problem). While testing on the PC, I decided one of the two models wasn't suitable there and said "delete it," meaning the copy on the PC. But I said that in what I thought was the PC session — it was actually the Mac session. The model wasn't a bad fit for the Mac at all; the mistake was purely about which machine's files the delete command was about to hit. I only caught it because I happened to double-check which machine I was in before the delete actually ran.

Proposed Solution

Show a persistent, system-enforced visual indicator (icon and/or hostname) on every session tab/row that identifies which physical machine it targets — not just as part of an editable text title, but as a fixed UI element the user can't accidentally overwrite or omit. OpenAI's Codex CLI already does something like this (it shows a globe-style icon next to sessions running on a remote/different machine), and it makes cross-machine context immediately obvious without having to read the title text carefully every time.

Alternative Solutions

Currently the only workaround is manually prefixing the session title with the machine name and hoping I remember to check it every time before running a command — which is exactly what failed in the incident described above.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

I have two large AI models downloaded on a Windows PC (RTX 5090), and a separate Claude Code session open on my Mac for unrelated work. I'm testing the models on the PC and decide one of them isn't suitable there, so I tell Claude "delete it," meaning the copy on the PC.

But I'm actually typing into the Mac session, not the PC session — both windows look the same, and nothing on screen tells me which machine I'm talking to. With a persistent machine indicator on the tab, I would have seen at a glance that I was in the Mac session and caught the mistake before confirming, instead of having to manually double-check.

Additional Context

_No response_