[Feature Request] Add timestamps to terminal interactions

Description:

I'd like to propose adding optional timestamps next to each interaction in the Claude Code terminal interface. This would display the date/time when each prompt was submitted and when Claude's responses were received.

Problem Statement:

Currently, when working with Claude Code for extended sessions or revisiting a terminal with an ongoing conversation, there's no way to see when interactions occurred. This makes it difficult to:

• Track how long tasks took to complete
• Understand the chronological flow of a debugging session
• Reference when specific changes were made during development
• Coordinate work across team members who might review the terminal output
• Create accurate development logs or time tracking records

Proposed Solution:

Add a toggleable timestamp feature that displays timestamps inline with each interaction. For example:

[2025-07-23 14:32:15] > Refactor the authentication module to use JWT tokens
[2025-07-23 14:32:18] I'll help you refactor the authentication module to use JWT tokens...

[2025-07-23 14:35:42] > Run the tests to make sure nothing broke
[2025-07-23 14:35:43] I'll run the tests to verify the refactoring...

Implementation Suggestions:

1. Toggle command: /timestamps [on|off] to enable/disable timestamps
2. Format options: /timestamps format [iso|relative|custom]

• ISO: 2025-07-23 14:32:15
• Relative: 2 minutes ago
• Custom: User-defined format string

1. Persistent setting: Save timestamp preference in .claude/settings.json
2. Color coding: Use subtle coloring to distinguish timestamps from content

Use Cases:

1. Development Documentation: Developers can copy terminal output with timestamps to document when specific decisions or changes were made
2. Debugging Sessions: When troubleshooting issues, timestamps help identify which attempts were made when and how long each approach took
3. Team Collaboration: When sharing terminal sessions or screenshots, timestamps provide context for when work was performed
4. Time Tracking: Freelancers and contractors can use timestamps to accurately bill for time spent on specific tasks
5. Learning & Training: When teaching others or creating tutorials, timestamps help demonstrate the realistic pace of development
6. Audit Trail: For compliance or review purposes, having timestamped records of AI-assisted development work
7. Performance Analysis: Understanding how long Claude takes to respond to different types of requests

Additional Benefits:

• Helps identify patterns in productivity (e.g., which times of day are most productive)
• Makes it easier to resume work after breaks by seeing exactly when you left off
• Provides context when reviewing old sessions to understand the development timeline
• Assists in creating more accurate commit messages by referencing when changes were actually made

Related Issues:

• This would complement #394 (view history of claude output from all sessions) by adding temporal context to interactions

Alternative Considered:
While the interaction logs contain timestamps in JSONL format, having them visible in real-time during the terminal session would be much more convenient and useful for the above use cases.