Feature request: Unified session history sync across VS Code extension, desktop app, and CLI

Summary

Currently, Claude Code sessions are siloed per client (VS Code extension, standalone desktop app, CLI), even when signed into the same account on the same machine. Sessions created in one surface don't appear in another, which makes it hard to continue work across environments.

Motivation / use case

• Same machine, same account, same project — but no shared history.
• I often start a conversation in the VS Code extension, then later want to continue it in the desktop app (or vice versa) and the chat is just missing.
• The previous related issue (#49775) is locked, so opening this one to keep the discussion alive.

Proposed behavior

• Sessions from ~/.claude/projects/<project-key>/*.jsonl should be visible across all official Claude Code surfaces signed into the same account.
• Ideally an opt-in cloud-sync option for cross-machine continuity as well, but local cross-surface visibility on the same machine would already be a huge win.

Workarounds today

• Stick to one surface per project
• Third-party tools (e.g., Claude Sync VS Code extension) to sync .claude/ via cloud storage — but unsupported and brittle

Environment

• OS: Windows 11
• Surfaces affected: VS Code extension + Claude Code Windows desktop app

Thanks!