[Feature Request] Seamless session sharing between Claude Code and Claude Desktop

Bug Description
Feature Request: Seamless Session Sharing between Claude Code and Claude Desktop

Problem:
Currently, there's no direct way to share conversation context between Claude Code (CLI) and Claude Desktop. Users must manually export to files, upload, or copy-paste - which breaks the flow of work.

Use Cases:

1. Code → Desktop: Preserve development discussions as permanent history (Code sessions feel volatile)
2. Desktop → Code: Continue research/planning directly into implementation

Requested Feature:

• One-click "Share to Desktop" / "Share to Code" button
• Or a shared conversation history accessible from both interfaces
• Ideally with selective sharing (share specific messages or summaries)

Why it matters:

• Claude Code excels at coding but sessions feel temporary
• Claude Desktop excels at research/brainstorming with persistent history
• Seamless handoff would create a powerful unified workflow

This would significantly improve the developer experience for users who rely on both tools daily.

Environment Info

• Platform: darwin
• Terminal: iTerm.app
• Version: 2.0.76
• Feedback ID: 9127bd17-ea9f-44e5-91b8-2998c1c30f90