[FEATURE] Conversation branching from sidebar (Claude Desktop) — fork/duplicate chat without losing main thread

Problem

In Claude Desktop / Claude.ai, conversations naturally branch off into sub-topics, but there is no first-class way to fork a chat as a separate, navigable thread. The only existing branching mechanism is implicit: editing an earlier user message creates a hidden branch that can only be reached via inline < > arrows below the message. This is:

• Invisible from the sidebar — branches don't appear as separate chats
• Easy to lose — once you keep typing on one branch, the other is hard to find again
• Not discoverable — most users don't know edit-message creates a branch
• Right-clicking a chat in the sidebar only offers Rename / Star / Delete, no Duplicate or Branch

This is purely an end-user / Claude Desktop concern, not a Claude Code CLI concern (existing issues #10370, #16236, #32631 are TUI/core-focused).

Proposed feature

Add a sidebar-level "Branch from here" or "Duplicate chat" action:

1. Right-click on a chat in the sidebar → Branch conversation
2. Right-click on any message inside a chat → Branch from this message (creates a new sidebar chat that inherits history up to that point)
3. The new branch appears as its own entry in the sidebar, optionally nested under the parent, so the main thread is never "lost"

Why this matters for end users

• Power users (lawyers, researchers, devs exploring designs) routinely want to explore "what if" alternatives without polluting the main thread
• The current edit-message workflow is a workaround, not a feature — it requires users to know an undocumented behavior
• ChatGPT has similar limitations; shipping a clean sidebar-level branching UX would be a clear differentiator

Related but distinct

• #10370, #16236, #16276, #32631 — all Claude Code (TUI) focused
• Side chat (Cmd+;) — useful but ephemeral, not persisted as a navigable branch in the sidebar