[FEATURE] Session continuity across devices — web and desktop should share session history

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Problem

Claude Code desktop sessions are not accessible from claude.ai/code (web) or other devices.
When switching to a laptop or mobile during a meeting, the entire session context is lost.

Use case

I was in a long productive coding session (6+ hours, building a full SaaS product).
Needed to access Claude mid-meeting from another device — session was completely unavailable.

The memory system partially mitigates this, but conversation history and accumulated
context are gone.

Expected behavior

Sessions should be accessible across devices when logged into the same account,
similar to how Cursor, VS Code with GitHub sync, or even ChatGPT handles conversation history.

Why it matters

This is a blocker for professional use — developers don't always work from a single machine.

Proposed Solution

Desenvolvam isso.

Alternative Solutions

Nao existe.

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

_No response_

Additional Context

_No response_