[FEATURE] Add /fork (conversation branching) support to VS Code 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

Summary

The CLI supports conversation forking via --fork-session (and /fork slash command), allowing users to branch a conversation at any point and explore a different direction while preserving the original session. This capability is not available in the VS Code extension — there is no slash command, hover action, command palette entry, or any other mechanism to fork a conversation.

Use case

When working on a complex task in the VS Code extension, I often reach a point where I want to explore an alternative approach without losing the current conversation context. Today the only option is to open a new chat (which starts fresh with no context) or switch to the terminal CLI to use --fork-session.

Conversation forking is particularly valuable for:

• Exploring alternative implementations — branch at a decision point, try both approaches
• Preserving working context — keep a "known good" conversation while experimenting
• A/B comparison — fork and compare Claude's output under different instructions

Expected behavior

• /fork slash command available in the VS Code extension prompt box (alongside existing commands like /compact, /usage, etc.)
• Forked session appears as a new tab, preserving full conversation history up to the fork point
• Parent session ID visible in the forked session (per #45655)

Current behavior

No fork/branch capability exists in the VS Code extension UI. The only workaround is opening a terminal and running claude --continue --fork-session.

Environment

• VS Code extension (macOS)
• CLI fork works correctly via --fork-session

Related issues

• #45655 — /fork should display parent session ID (CLI)
• #44684 — --fork-session forks last-flushed JSONL, not live conversation (CLI)

Proposed Solution

Expected behavior

• /fork slash command available in the VS Code extension prompt box (alongside existing commands like /compact, /usage, etc.)
• Forked session appears as a new tab, preserving full conversation history up to the fork point
• Parent session ID visible in the forked session (per #45655)

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

This is the same scenario as the existing fork feature, simply supported in the VS Code extension. The justifications are identical.

Additional Context

_No response_