[FEATURE] Add /fork (conversation branching) support to the Cursor extension

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

The CLI supports conversation forking via --fork-session (and the /fork slash command), allowing users to branch a conversation at any point and explore a different direction while preserving the original session.

The Claude Code extension running in Cursor is missing this: there is no /fork in the prompt box, so the only way to branch is to scroll back and fork from an earlier message — which loses all the context accumulated since that message. #69272 requests the same capability for the VS Code extension; this issue covers the Cursor extension specifically.

Proposed Solution

• /fork slash command available in the Cursor extension prompt box (alongside existing commands like /compact, /usage, etc.), forking from the current point of the conversation.
• The forked session appears as a new tab, preserving the full conversation history up to the fork point.

Alternative Solutions

Currently the only workaround is to leave the extension and resume the session in the terminal with claude --fork-session, which drops the IDE integration.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. Mid-session, Claude proposes an approach and I want to try two directions from the same context.
2. In the CLI I would /fork and explore one direction per session.
3. In Cursor I cannot: I either overwrite the current direction or fork from an old message and lose everything since.

Additional Context

VS Code counterpart: #69272. Same behavior as requested there, for the Cursor extension.