[FEATURE] Desktop app: save a pane layout as the default for new sessions

Problem

In the Code tab of the Claude desktop app (macOS), I set up each session's side panes the same way: terminal, background tasks and the built-in browser, each in a specific spot. Every new session opens with the stock layout, so I have to put the panes back by hand every time. No setting saves a layout, and the docs do not mention one.

Proposed solution

• A "Save current layout as default" action, from the View menu or a pane header menu.
• New sessions open with the saved layout: which panes are open, where each one sits, and their sizes.
• A "Reset to stock layout" action.
• Nice to have: popped-out pane windows are part of the saved layout and reopen in the same spot (related: #98335, the pop-out control is currently missing).

Why it matters

Anyone on more than one monitor, or anyone who always works with the same panes, rebuilds the same layout many times a day.

Environment

Claude desktop app, Code tab, macOS 15 (Darwin 24.6).