[FEATURE] Per-session status indicator (colored dot) in the VS Code extension's session list

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 working with several concurrent sessions in the Claude Code VS Code extension, the Sessions panel (and the editor tabs) don't show each session's current state at a glance. To find out whether a session is thinking, streaming a response, waiting for my permission/input, finished, or errored, I have to click into each one.

With multiple sessions open this makes it easy to miss a session that is blocked waiting on a permission prompt, or to not notice which sessions have finished. Sounds via Notification/Stop hooks only fire for the focused/visible session, so background tabs give no cue.

Proposed Solution

Show a small colored status dot next to each session in the extension's Sessions panel (and ideally on the editor tab), color-coded by state — for example:

• 🟢 green — idle / finished (waiting for the user)
• 🔵 blue (pulsing) — running / streaming the answer
• 🟣 purple (pulsing) — thinking / extended reasoning
• 🟡 amber (blinking) — waiting for permission or input (needs attention)
• 🔴 red — last turn ended with an error

This lets me monitor many sessions at a glance and immediately spot the one that needs attention, without clicking through each session.

Alternative Solutions

• Notification/Stop hooks (sounds): work, but only for the focused/visible session — background tabs don't fire the hook.
• OS / mobile push (inputNeededNotifEnabled): helps for "needs input" but is heavier and not an at-a-glance visual across all open sessions.
• Custom statusLine: appears to be CLI/terminal-only; it does not render in the VS Code extension panel.

None of these provide an at-a-glance, per-session visual state inside the extension UI.

Additional Context

The underlying state already exists internally (running / waiting-permission / idle / error, and streaming-thinking vs streaming-text can distinguish "thinking" from "writing"). A colored dot in the Sessions list would just surface it.

I prototyped exactly this in a third-party multi-session manager built on the Agent SDK and it works well in practice — would love to have it natively in the extension where most of my day-to-day work happens.

_(Filing as an enhancement.)_