Feature request: portable conversation context across Claude Code, claude.ai, and Cowork

Summary

A conversation started in one Claude surface can't be opened or continued in another. There's no way to carry conversation context across Claude Code, claude.ai, and Cowork.

Current behavior

• A claude.ai/share/... conversation link is not readable from Claude Code (WebFetch hits a login wall / SPA shell).
• claude.ai Projects are sandboxed — their conversation/context state isn't reachable from Claude Code, and vice versa.
• The only working bridge today is shared files + MCP (e.g. a shared Google Drive / Obsidian vault, or memory/task MCP servers). That moves artifacts, not the conversation.

Why this matters

A common workflow is to plan/strategize on a tablet via claude.ai and build on a laptop via Claude Code. Today that handoff loses the conversation entirely — you re-establish context by hand. As competitors consolidate code + chat into a single app, this cross-surface discontinuity is a concrete friction point and a competitive disadvantage.

Requested behavior

A way to continue the same conversation across surfaces — e.g. open a claude.ai conversation (or Project thread) inside Claude Code and keep going, and vice versa — so context is portable rather than re-created manually.

Workaround

Persist shared state to files + MCP (BrianPKM/Obsidian vault, memory, dashboard tasks) so each surface can re-read it. Works for documents and structured state; does not carry the conversation thread itself.