Feature request: display timestamps on messages in conversation

Problem

There is no way to see when messages were sent or received in a Claude Code conversation — neither user prompts nor assistant responses display timestamps. This applies to all three clients: CLI (terminal), IDE extensions (VS Code / JetBrains), and the web app (claude.ai/code).

Why timestamps matter

When working on multi-hour sessions or reviewing a conversation after the fact, temporal context is critical:

• Referencing past interactions: "the thing we discussed this morning" has no anchor point without timestamps
• Debugging agent behavior: when investigating autonomous agent issues (cron jobs, scheduled tasks), correlating conversation events with external logs requires knowing when each exchange happened
• Multi-device workflows: users who switch between terminal, phone, and IDE throughout the day lose all sense of timeline
• Context for AI: if the AI could see when messages were sent, it could better reason about recency, urgency, and temporal relationships between events the user describes

Workarounds considered

• CLI hook (user-prompt-submit): can log timestamps to a file, but doesn't display them inline in the conversation — and doesn't cover IDE or web clients
• Prefixing messages manually: impractical and doesn't timestamp the AI's responses
• No config option exists across any of the three clients to enable this

Proposal

Display a subtle timestamp (e.g., 14:32 or 2:32 PM depending on locale) next to each message — both user prompts and assistant responses. This is standard in virtually every chat interface (Slack, Discord, iMessage, etc.) and its absence in Claude Code is a notable gap.

Ideally configurable (show/hide) for users who prefer a clean view.