[FEATURE] Bring the Code tab's side chat (/btw, Ctrl+;) to the Chat and Cowork tabs

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

In long Chat / Cowork conversations there is no way to ask "wait, I didn't understand that part — can you rephrase?" without consequences: the clarifying exchange scrolls the main thread away and pollutes context that is still needed for the ongoing work.

Current workarounds are all clumsy:

• Open a second conversation and manually copy-paste the confusing fragment — the second chat can't see the first one, so you re-explain everything.
• Ask inline — the answer you were still reading scrolls away, and the conversation fills with meta-discussion noise.

The Code tab of the desktop app already solves exactly this with side chat (Cmd+; / Ctrl+;, or /btw):

"A side chat lets you ask Claude a question that uses your session's context but doesn't add anything back to the main conversation." — https://code.claude.com/docs/en/desktop

But side chat exists only in the Code tab. Chat and Cowork — the tabs where long, reading-heavy conversations actually happen — don't have it.

Proposed Solution

Port the existing side chat to the Chat and Cowork tabs of the desktop app (and ideally claude.ai web):

• Same shortcut (Ctrl+; / Cmd+;) opens a side panel next to the conversation
• The side chat can read the full conversation up to that point
• Side-chat messages never enter the main conversation's context
• Close to discard

Use cases:

• Ask to rephrase / clarify part of an answer while the main thread stays scrolled exactly where it was
• Quick "what does this term mean?" without derailing a working conversation
• In Cowork: ask about a running task's output without injecting noise into the agent's session context

This is tab parity rather than a new feature — the UX, naming, shortcut, and context model are already designed, shipped, and documented for the Code tab.

Related: #60980 (minimize side-chat panels, Code tab), #26716 (native quote UI), #59029 (conversation branching in Desktop), #48099 (TUI side chat from a text selection, closed as not planned — this request is narrower: make the existing GUI side chat available in more tabs).

Alternative Solutions

• A second conversation in another window + manual copy-paste of the confusing fragment — the second chat can't see the first one, so context is re-explained by hand every time.
• Asking Claude to maintain a running "explanations" artifact in the side panel — partially works, but the clarifying exchange still lands in the main thread and consumes its context.
• In the Code tab this is already a solved problem (/btw) — but Chat/Cowork conversations can't be moved there.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

Example scenario:

1. In a long Cowork session, Claude produces a multi-paragraph technical answer (e.g. a step-by-step infrastructure debugging plan).
2. Halfway through reading it, I hit a term or a step I don't understand.
3. Today I either ask inline (the plan scrolls away and the thread fills with clarification noise) or open a second conversation and manually re-paste the fragment plus enough context for it to make sense.
4. With this feature: Ctrl+; opens a side chat that already sees the whole thread; I ask "rephrase step 3 in simpler terms", read the answer, close the panel.
5. The main conversation is untouched — same scroll position, same context, ready to continue.

Additional Context

_No response_