Tabbed conversations in VS Code extension panel

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

When working with multiple Claude Code conversations in the VS Code extension, the only way to switch between them is through the "Session history" dropdown. This makes it hard to quickly jump between active conversations, especially when working on different tasks in parallel.

This is distinct from #10747 (rename/delete chats) — that issue covers chat management, while this request is about the navigation UX for active conversations.

Proposed Solution

Add a tab bar at the top of the Claude Code panel (next to the conversation title) showing all open/recent conversations as clickable tabs — similar to how VS Code editor tabs work. Users could:

• See multiple conversations as tabs in the panel header
• Click a tab to instantly switch to that conversation
• Close a tab to dismiss it from the tab bar (without deleting the conversation)
• Optionally drag to reorder tabs

Alternative Solutions

• Opening multiple Claude Code panels side by side (current workaround, but uses more screen space)
• A pinned/favorites list in the past conversations dropdown

Priority

Nice to have

Feature Category

IDE integration

Use Case Example

Working on a feature in one conversation while debugging a separate issue in another. Currently requires navigating through the past conversations menu each time, losing flow.

Additional Context

Related but distinct from #10747 which covers rename/delete. Multiple issues requesting tabs (#24377, #26135, #11145) were closed as duplicates of #10747, but the tabbed UI concept is a separate UX improvement from chat management.