[FEATURE] Detach the Browser pane into its own window (multi-monitor)

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

Claude drives a browser-rendered layout and takes measurements while I watch
the drawing and judge whether it looks right. Sharing one window between the transcript
and the rendered page means constantly trading space between the two — and since I
approve every browser call by hand, the pane is in near-continuous use.

Proposed Solution

The Browser pane is docked inside the Claude Code window. I'd like to pull the pane
itself out as a separate OS window and park it on a second monitor.

Not covered by the existing requests: #45206, #27725, #50963 and #30154 all detach at
the session/tab level. Detaching a session doesn't help here, because the Browser pane
travels inside that window and still shares space with the transcript. What I want out
is the pane.

Windows 11, desktop app.

Alternative Solutions

The Chrome extension isn't a substitute — it solves placement by using my own browser,
at the cost of the built-in pane's isolation and my real profile (see also #87764).

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

_No response_

Additional Context

_No response_