[FEATURE] Show background agent output in separate tab

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

I'm using background agents in VS Code to do things like code reviews and testing. All the output from these agents is streamed into my Clause Code tab making it really hard to descern agent output from my foreground activity.

Proposed Solution

I'd like to have the Claude Code extension for VS Code open a separate tab for each background agent and stream the agent's output into that tab. I'm fine with manually closing the background agent tab after the agent is done.

Alternative Solutions

An alternative is to make the color of background agent output different than output from the foreground interactions.

Priority

High - Significant impact on productivity

Feature Category

Performance and speed

Use Case Example

1. I'm working with Claude Code in VS Code
2. Claude has finished fixing a bug and envokes a code review agent to run in the background while updating the product documentation in the foreground

Additional Context

_No response_