[FEATURE] Running/idle visual indicator on session tabs in VSCode extension

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

When multiple Claude Code sessions are open in the VSCode extension panel, I can't tell which one is actively running and which has finished without clicking into each tab. The session tabs only show names — no status indicator.

Proposed Solution

Suggested solution

Add a subtle visual indicator on each session tab:

• A small pulsing/animated dot (e.g. green when running, grey/static when idle)
• Or a color accent on the tab itself

This is common in terminal emulators (Warp, iTerm2 plugins) and would save a lot of tab-switching.

Alternative Solutions

• Activity Bar spark icon already has blue dot (permission pending) and orange dot (finished while hidden), but neither covers the "currently running" state
• Terminal-based workarounds (claude-code-tab-title, Claudelike-Bar) don't work with the native VSCode extension panel
• Hooks can run shell commands but can't communicate back to the webview UI

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

1. I have Claude Code sessions open for backend API work and frontend UI work simultaneously
2. I ask the backend session to refactor a module — it starts running
3. I switch to the frontend tab to work on something else while waiting
4. Without looking at each tab, I have no idea the backend session has finished and is waiting for my next input
5. With a running/idle indicator, I'd see the dot change from pulsing green to static grey and know immediately
6. This saves me from constantly cycling through tabs to check "are you done yet?"

Additional Context

This is purely a UI enhancement within Anthropic's own webview panel — no VSCode API limitations involved.