[FEATURE] chat + Code tab side-by-side

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

The Claude Desktop app's Code tab supports split view for running multiple Code sessions side by side (via Cmd/Option-click, see #56278). However, there is no way to view a regular Chat conversation alongside a Code session in the same window. A previous request for Chat + Cowork split view (#29136) was closed as not related to Claude Code, since Cowork and Code are different products — this request is scoped specifically to Chat + Code.

Users who think through a problem conversationally in Chat while actively steering a Claude Code session (e.g. via Remote Control) currently have to tab-switch between Chat and Code, losing visual continuity on whichever panel they leave.

### Proposed Solution

Extend the existing Code tab split-view mechanism to also accept a Chat conversation as one of the panes — for example, by allowing a Chat conversation to be dragged into a Code tab split, or by adding a "pin as side panel" option from within Chat that keeps it visible while a Code session runs in the adjacent pane.

### Alternative Solutions

Currently i work around this by using the desktop app for a code session and a browser for the chat.

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. A Code session is running a long task on a remote machine via Remote Control.
2. The user wants to think through next steps or draft a message in a regular Chat conversation without losing sight of the Code session's progress.
3. Currently this requires switching tabs; with this feature, both would be visible in the same window.

### Additional Context

Related but distinct from #29136 (Chat + Cowork, closed as invalid/unrelated) and #48688 (split panes within Code tab only).