[FEATURE] VS Code extension: session-list status indicators + notifications when a background session finishes or needs attention

Environment

• Claude Code VS Code extension 2.1.212 (native panel UI, not integrated terminal)
• VS Code on Windows connected via Remote-SSH to a Linux host (Claude Code runs on the remote)
• Workflow: several parallel sessions open as editor tabs, plus the extension's session list in the sidebar

Problem

When running multiple sessions in parallel, there is no way to notice that a background session finished its turn or is waiting for input/permission without manually clicking through every tab:

• The session list in the sidebar is static — it only shows the session title and relative time. No spinner for a running session, no dot/badge for one that needs attention.
• The colored dot indicator only appears on hidden editor tabs, which is easy to miss and not visible at all when the tab bar is crowded or when working in another editor group.
• The extension sends no OS/VS Code notifications when a session ends its turn or hits a permission prompt. preferredNotifChannel appears to be CLI-only and has no effect in the extension.
• In the Remote-SSH scenario, Notification/Stop hooks run on the remote Linux host, so desktop-notification hooks (notify-send etc.) can't reach the local Windows desktop either. The extension UI is the only component that lives on the local machine — so it's the only place that could reliably alert the user.

The result: with 3–5 parallel sessions, finished or blocked sessions silently sit there and the user has to poll tabs by hand.

Feature request

1. Live status in the session list: spinner while a session is working, a distinct needs-attention indicator (permission prompt / waiting for input), and a finished marker — so the sidebar list works as a glanceable dashboard for parallel sessions.
2. Optional notifications from the extension when a session finishes its turn or needs permission — e.g. via the VS Code notifications API (toast + bell), which would work in Remote-SSH setups too since the VS Code UI runs locally. A setting to opt in/out (and ideally filter: permission-needed only vs. all turn-ends) would cover both quiet and noisy preferences.

Either of the two would already remove most of the pain; together they'd make parallel-session workflows genuinely usable in the extension.