[FEATURE] Emacs/tmux-style keyboard pane splitting (split-right / split-down) in desktop Code tab; new sessions shouldn't replace an open pane

Preflight Checklist

• [x] I have searched existing requests — the closest are all closed/stale and none cover this exact gap (see Additional Context).
• [x] This is a single feature request.

Problem Statement

In the desktop app's Code tab, the only way to view two sessions side-by-side is the Cmd-click gesture on a sidebar session — which is undocumented (#56278) and only ever splits to the right.

The bigger pain: once you have panes open, pressing Cmd-N (or clicking + New session) replaces one of the existing panes with the new chat instead of creating a new pane. So starting a new session inevitably clobbers a window you were already using. There is also no way to split downward, and no keyboard shortcut to split at all.

Coming from Emacs / tmux / IDE split editors, the expectation is that splitting is an explicit, keyboard-driven action — and that opening something new lands in a new pane, not on top of an existing one.

Proposed Solution

Add Emacs/tmux-style pane management to the Code tab, driven by keyboard shortcuts:

• Split right — open a new pane to the right of the focused pane (e.g. Cmd-D, à la iTerm; or C-x 3 for the Emacs crowd).
• Split down — open a new pane below the focused pane (e.g. Cmd-Shift-D; or C-x 2).
• New session into a new pane — Cmd-N / + New session should open into a new split (or a clearly designated pane), never silently overwrite another pane that's in use.
• Move focus between panes by keyboard (e.g. Cmd-[ / Cmd-], or C-x o).
• Close the focused pane by keyboard (e.g. Cmd-W scoped to the pane, or C-x 0).

Alternative Solutions

At minimum: make Cmd-N / + New session open a new pane instead of replacing an existing one, and document the existing Cmd-click split gesture. Even without full keyboard split bindings, not clobbering an active pane would remove most of the friction.

Priority

Medium — Would be very helpful for anyone running multiple sessions at once.

Feature Category

Desktop app / UI

Use Case Example

Reading context or instructions in one session while driving a second; running several agents in parallel and glancing between them; comparing two conversations — all with the same muscle memory as tmux/Emacs/VS Code split editors, without a mouse and without losing a pane every time a new chat starts.

Additional Context

Related but distinct prior issues (all currently closed/stale):

• #48944 — Resizable & vertical 3-column split view panels — about resizing/arranging panes that are already open, not about how to create splits.
• #56278 — documents the existing Cmd-click → split-right gesture (undocumented).
• #59038 — a regression report that the Cmd-click split stopped working.

None of these cover keyboard-driven split-right / split-down or the "new session replaces an existing pane" behavior, which is the core of this request.