Feature: Push notifications for completed tasks in remote-control mode

Feature Request

When using Claude Code in remote-control mode via the Claude app, it would be great to receive push notifications when a task completes while the app is backgrounded.

Use Case

Running multiple Claude Code sessions via tmux with remote-control. Tasks can take a while to complete, and there's no way to know when they finish without switching back to the app and checking manually.

A push notification (iOS/Android/desktop) when the remote-controlled session finishes a task would make the workflow much smoother — especially when multitasking or running several sessions in parallel.