[FEATURE] Browser notification when task completes in Claude.ai web version

Feature Request

Problem

When using Claude Code via the claude.ai web interface (remote sessions), there is no notification when a long-running task completes. Users often switch to other browser tabs while waiting for tasks to finish, but have no way of knowing when Claude is done and waiting for input.

Proposed Solution

Implement browser notifications (using the Web Notifications API) to alert users when:

• A task/response completes
• Claude is waiting for user input
• Background agents finish their work

Expected Behavior

1. User grants notification permission (one-time browser prompt)
2. User switches to another tab while Claude works
3. When Claude completes the task, a browser notification appears
4. Clicking the notification focuses the Claude.ai tab

Additional Context

• The CLI version already has terminal bell/notification support
• This would significantly improve UX for users who multitask while waiting for Claude
• Similar to how GitHub Actions notifies when workflows complete

Environment

• Platform: Web (claude.ai)
• Feature area: claude-code-web, tui