Feature Request: Sync CLI conversation history with Claude desktop app

Summary

It would be great if conversations from the Claude Code CLI were synced and visible in the Claude desktop app, similar to how OpenAI's Codex CLI syncs conversations with the ChatGPT desktop app.

Motivation

The desktop app provides a much more comfortable UI for browsing and reviewing past conversations — search, sidebar navigation, timestamps, etc. Having to choose between the power of the CLI and the convenience of the desktop UI is an unnecessary tradeoff.

Expected behavior

• Conversations started in claude CLI are reflected in the Claude desktop app's conversation list
• Users can review past sessions from either interface
• Ideally, the sync is account-based (similar to Codex ↔ ChatGPT desktop)

Current behavior

CLI sessions are local-only and not visible in the Claude desktop app. The two products are completely isolated.

Alternatives considered

• Manually exporting conversation history (not practical for daily use)
• Using only the desktop app (loses the power-user CLI workflow)