[FEATURE] Display Message Timestamps in the Chat Panel

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

There is no way to see when a message was sent in the Claude Code VS Code extension chat panel. Long or multi-session conversations have no per-message timestamps, making it impossible to correlate AI responses with external events (code changes, build failures, etc.) or to audit how long a task took.

Proposed Solution

Show a subtle timestamp beside or below each message in the chat panel (e.g., display on hover, similar to Slack or VS Code's built-in Chat view). Optionally add a setting to control visibility: "always" | "hover" | "never".

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

_No response_

Additional Context

_No response_