[FEATURE] Show wall-clock timestamps on each message/tool call row in chat UI

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

Add an option to display wall-clock timestamps (e.g. 14:32:05) next to each message and tool call row in the Claude Code chat interface (VSCode extension and CLI).

Proposed Solution

Add an optional setting (e.g. showTimestamps: true) that, when enabled, renders a wall-clock timestamp next to each message bubble and tool call row in the chat UI.

• Format: HH:mm:ss or HH:mm:ss.SSS (configurable, or locale-aware)
• Applies to: user messages, assistant messages, and individual tool call rows (the "IN" / "OUT" blocks in the VSCode extension)
• Should work in both the VSCode extension and the terminal CLI

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_