[Feature Request] Add per-message timestamp option to CLI output

Bug Description
Feature Request: Per-message timestamp option

Would love a setting to prefix each Claude Code output message with a high-precision timestamp, e.g.:
2026-02-25 15:55:20.012 Done. The new diff covers all changes...

This could be customizable if possible.

Use cases:

• Session auditing and debugging
• Time-tracking how long each response takes
• Correlating Claude output with external logs
• Reviewing session transcripts with temporal context

Proposed implementation (any of these would work):

1. CLI flag: claude --show-timestamps
2. settings.json option: { "showTimestamps": true, "timestampFormat": "yyyy-MM-dd HH:mm:ss.SSS" }
3. Environment variable: CLAUDE_SHOW_TIMESTAMPS=1

The transcript .jsonl already has timestamps per API call — this would just surface them in the terminal output. Current workarounds (terminal pipe, post-processing .jsonl) break interactive features or are after-the-fact only.

Environment Info

• Platform: darwin
• Terminal: iTerm.app
• Version: 2.1.50
• Feedback ID: aa2c9f23-4d03-444d-8fbf-1c7cceb54fcb