[FEATURE] Add --worktree support for parallel sessions in VS Code (same as Agent View in VSCode does)

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 VSCode plugin for Claude is not yet supporting the option to create worktrees.
Please add VS Code extension: Add --worktree support for parallel sessions

Allow each Claude Code conversation tab/window in VS Code to run in a separate Git worktree, like the CLI does with claude --worktree <name>. This enables multiple agents to work on the same codebase simultaneously without conflicts, and create separate PRs per feature.

Currently all VS Code sessions operate on the same main branch. The CLI already supports this perfectly—the extension just needs to expose it.

Proposed Solution

Be able to choose for each session in VSCode Claude if it should work in a separate worktree or not.
And show when a session is completed or still processing.

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

Interactive mode (TUI)

Use Case Example

_No response_

Additional Context

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