[Feature Request] Add task completion notifications for Claude Code web version

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

When running Claude Code in a Chrome browser, I often switch to other tabs or applications while waiting for long-running tasks to complete. Without any notification system, I have to constantly check back to see if the task has finished.

Key observation: The regular Claude AI chat (claude.ai) already supports browser notifications when responses are ready, and this works great. However, Claude Code's web version does not have this feature implemented yet.

This is particularly frustrating when:

• Running complex code generation or refactoring tasks that take several minutes
• Working on multiple projects and need to context-switch efficiently
• The task completes quickly but I've already moved to another tab

Currently, there's no way to know when Claude Code has finished processing without actively monitoring the tab, unlike the main Claude AI interface which notifies users properly.

Proposed Solution

I'd like Claude Code's web version to implement task completion notifications, similar to how the main Claude AI chat (claude.ai) already does.

Reference implementation: The regular Claude AI chat already sends Chrome browser notifications when responses are complete. This same functionality should be extended to Claude Code's web interface.

Suggested implementation options (consistent with existing Claude AI behavior):

1. Browser notifications - Use the Web Notifications API to show a desktop notification when a task completes (same as claude.ai)
2. Tab title update - Change the browser tab title to indicate completion (e.g., "✓ Task Complete - Claude Code")
3. Sound notification - Play a subtle audio cue when tasks finish (with an option to disable)
4. Favicon badge - Update the favicon to show a completion indicator

A settings option to enable/disable these notifications would be helpful for users who prefer different workflows.

Alternative Solutions

Currently, I work around this by:

• Keeping the Claude Code tab visible on a secondary monitor
• Periodically checking the tab to see if processing has completed
• Using browser extensions that monitor page changes (not ideal)

Important comparison: The main Claude AI chat (claude.ai) already solves this perfectly with Chrome browser notifications. When I'm using regular Claude AI and switch tabs, I receive a notification when Claude's response is ready. This works great!

However, Claude Code's web version lacks this feature entirely, creating an inconsistent user experience across Anthropic's products. Since the notification infrastructure already exists in claude.ai, it should be relatively straightforward to implement the same functionality in Claude Code.

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

Example scenario:

1. I'm working on a large codebase refactoring task using Claude Code in my browser
2. I ask Claude Code to analyze and refactor multiple files, which takes 3-5 minutes
3. While waiting, I switch to another browser tab to review documentation or respond to Slack messages
4. With this feature, I would receive a browser notification saying "Claude Code: Task Complete" when the refactoring is done
5. This would save me time because I wouldn't need to constantly switch tabs to check progress, and I could immediately review the results when ready

Additional Context

_No response_