VS Code: Support tabbed conversations within a single panel

Feature Request

Is your feature request related to a problem?

When working with multiple Claude Code conversations in VS Code, each conversation opens as a separate editor tab. This clutters the editor tab bar and makes it harder to flip between conversations quickly.

In contrast, VS Code's integrated terminal supports multiple tabs within a single panel — you can easily flip between terminal sessions without them mixing into the editor tabs.

Describe the solution you'd like

Support tabbed conversations within a single Claude Code panel, similar to how terminal tabs work in VS Code. Multiple conversations would be grouped together in one panel with tabs to switch between them, rather than requiring separate editor tabs.

Describe alternatives you've considered

• Opening conversations as separate editor tabs — works but clutters the tab bar and mixes Claude tabs with code file tabs
• Using split editor views — takes up too much screen space when you just want to flip between conversations
• Dragging tabs to the bottom panel — VS Code doesn't allow this for webview-based extensions

Additional context

This would make the multi-conversation workflow feel much more natural, especially for users who are already accustomed to the terminal tab pattern in VS Code.