Claude App: Show notification dot when background task completes

Feature Request

When using the Claude app (not the CLI), there's no visual indicator when a long-running task completes if the app window is not in focus or the user has navigated away.

Desired Behavior

When a task completes in the Claude app, show a notification dot (badge) on the app icon / tab / conversation to indicate that results are ready. This is similar to how chat apps show unread message indicators.

Use Case

When working with Claude on multi-step coding tasks or long-running operations, users often switch to other windows/apps while waiting. Without a notification dot, there's no way to know the task has finished without manually checking back, which leads to wasted time either:

• Checking too frequently (interrupting other work)
• Checking too late (delaying the workflow)

Additional Context

This would significantly improve the multitasking workflow for users who rely on Claude for iterative development tasks. Most productivity apps (Slack, Discord, VS Code, etc.) provide similar notification badges when background activity completes.