[FEATURE] Show timestamps on chat messages in the VS Code 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

Description:

It would be helpful to show a timestamp next to each message in the Claude Code chat window (similar to how terminal sessions can display timestamps). This would make it easier to track when decisions were made, when commands were run, and how long tasks took — especially during long sessions.

Proposed Solution

Suggested behaviour:

Show a small timestamp (e.g. 14:32) next to each user and assistant message
Could be toggled on/off in settings

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

CLI commands and flags

Use Case Example

Use case:

When working on extended sessions (e.g. running background scripts, making multiple commits), it's useful to know at a glance when something was discussed or completed without having to cross-reference git logs or terminal history.

Additional Context

_No response_