[FEATURE] Dockable side chat: embed it as a resizable right-hand panel in the desktop app instead of a floating window

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

On the Claude Code desktop app (Windows/macOS), opening a side chat spawns a separate floating window that overlaps the main window. In practice this:

• Steals keyboard focus and covers the main conversation I'm working in.
• Breaks my flow: I have to manage a second OS window instead of keeping the side chat and my main work visible side by side.
• Is awkward on a multi-monitor setup — the window pops up where I don't want it and I keep repositioning it, or it slips behind the main window.

What makes this stand out: the desktop app already docks a panel on the right side (the Background Tasks / workflow panel). So the floating side chat is inconsistent with the app's own layout — there's already a docked right-hand region, but the side chat won't live there.

Proposed Solution

Let the side chat dock as a persistent, resizable panel on the right side of the main window, replacing the floating-window behavior.

Concretely:

• Docks to the right edge of the main window, next to the main conversation — the same region where the Background Tasks panel already docks.
• Freely resizable by dragging the divider between the main area and the panel.
• Collapsible to a thin strip / icon so I can reclaim space without closing the chat.
• A keyboard shortcut to show/hide the panel (a sensible default at the team's discretion, ideally rebindable).
• A single docked side chat is enough — no need for tabs or multiple side chats.
• It's acceptable for opening the panel to temporarily narrow or hide other panels (file tree / diff).
• The side chat keeps sharing context with the main conversation (working directory, open files, session), as it does today.
• Persistence can be minimal: remembering the last panel width is enough.

In short: keep the side chat in view next to my main work as a docked panel, instead of floating in a separate window.

Alternative Solutions

• Current workaround: manually move and resize the floating window every time, and chase it across monitors. That friction is exactly what this request is about.
• Dock/undock toggle: instead of fully replacing the floating window, the app could let users switch a side chat between docked-panel and floating-window modes. That would also satisfy #60108 (a floating side-chat option for streamers) — the opposite default, but compatible with the same toggle.

Related issues I found (none asks to dock the side chat as a right-hand panel, so this isn't a duplicate):

• #60108 — Floating window option for side-chats (opposite direction; open, marked stale).
• #45206 — Detachable/undockable Chat, Cowork, Code tabs for multi-monitor (about the main tabs, not the side chat; closed as duplicate).
• #29136 — Split view Chat + Cowork (not about side-chats; closed).

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. I'm running a multi-agent analysis in my main conversation (several agents in parallel plus a synthesis step).
2. While that runs, I open a side chat to ask a quick parallel question, debug something, or compare two answers — without polluting the main thread.
3. Today the side chat opens as a floating window that lands on top of my work, steals focus, and (on my dual-monitor setup) appears where I don't want it. I either reposition it or lose it behind the main window.
4. With a docked side panel, the side chat sits in a fixed, resizable column on the right, next to the main conversation. I can read both at once, resize it to taste, and collapse it when I need the space.
5. This keeps me in flow and matches how the desktop app already docks the Background Tasks panel on the right.

Additional Context

Mockup (current behavior, annotated):


Notes:

• In the mockup, the "Chat lateral" (side chat) is a separate floating window in the bottom-right, overlapping the main window (green arrow). At the same time, the app already shows a docked Background Tasks panel on the right — the side chat should be able to dock the same way.
• Prior art: IDE assistants (VS Code, JetBrains) and browser dev tools dock their side panels to a resizable edge column instead of using floating windows.
• Platform scope: this request targets the desktop app (Windows/macOS).

<img width="934" height="510" alt="Image" src="https://github.com/user-attachments/assets/71411184-5ec9-4788-81fc-320449052440" />