#400 feat(desktop): add path and branch info to workspace header

Summary

• Display worktree path with folder icon in workspace header
• Add searchable branch dropdown with pagination (10 branches per page)
• Show relative time since last commit for each branch (e.g., 2h, 3d)
• Mark checked-out branches as disabled with "(in use)" label
• Sort branches: current first, then checked-out, then by recent commit
• Add switchBranch tRPC mutation to change branches within worktree
• Extract formatRelativeTime utility with tests

Test plan

• Verify path displays correctly with folder icon
• Verify branch dropdown shows current branch
• Test search functionality filters branches correctly
• Test "Show more" pagination loads additional branches
• Verify checked-out branches are disabled and show "(in use)"
• Test branch switching works and updates UI
• Run bun test formatRelativeTime to verify utility tests pass

🤖 Generated with Claude Code

Summary by CodeRabbit

• New Features

  • Branch switching workflow added (switch branches per worktree) and interactive branch picker with search, pagination, checked-out indicators, and last-commit times.
  • Workspace header refreshed: masked path display, clickable "open in app" action, and branch selector moved above the workspace.

• Tests

  • Deterministic test suite added for relative-time formatting.

• Chores

  • Workspace VSCode settings/recommendations cleaned up and .gitignore compacted.

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