Show host machine name in session UI for remote sessions

Problem

When using the Claude Code desktop app to connect to remote/headless sessions, there is no visible indicator of which machine the session is actually executing on. The only way to know is to manually rename the session (e.g., "mac-mini-work") or run hostname.

This is particularly frustrating for users who work across multiple machines. Without a clear indicator, it's easy to lose track of which machine you're connected to — and potentially edit files on the wrong machine without realizing it.

Suggested solution

Surface the hostname (or a user-friendly machine name) somewhere persistent in the session UI — for example in the session header, sidebar, or status bar. This would make it immediately clear whether you're connected to cliffs-mac-mini.local vs cliffs-macbook-pro.local without having to rely on manual naming conventions.

Context

• Using Claude Code desktop app on macOS
• Running headless sessions on one machine (Mac Mini), connecting from another (MacBook Pro)
• The session list and UI give no indication of which host is being controlled

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