[Extension] Tab management: side-by-side tabs, rename, and close conversations

Feature Request

The VS Code extension (v2.1.37) lacks basic conversation management features that exist in other AI coding tools (e.g. Cursor Agent).

Missing features:

1. Side-by-side conversation tabs — Currently conversations are only accessible via a dropdown list. Users running multiple agents in parallel need to see them side-by-side, like editor tabs.

1. Rename conversations — No way to rename a chat. The first prompt becomes the title, which is often not descriptive enough.

1. Close/delete conversations — No way to close or remove old conversations from the list. They accumulate with no management option.

Why it matters:

For developers working on large codebases (170K+ lines) who run multiple agents on different tasks simultaneously, conversation management is essential for productivity. The current dropdown-only navigation slows down parallel workflows significantly.

Environment:

• Claude Code Extension v2.1.37
• Cursor IDE (VS Code based)
• macOS