Feature request: Share conversation history between CLI and desktop app

Problem

The Claude Code CLI and desktop app maintain completely separate session states with no cross-client synchronization. When switching between the two clients, all conversation context is lost.

Proposed Solution

Sync conversation history across clients (CLI ↔ desktop app) so users can pick up where they left off regardless of which client they're using.

Use Case

Working on a project via CLI, then continuing the same session in the desktop app (or vice versa) without losing context.

Current Workaround

The file-based memory system (~/.claude/projects/.../memory/) partially addresses this by persisting structured key facts across sessions, but it doesn't preserve full conversation history.

Additional Context

This would be especially useful for long-running projects where conversation context carries significant value.