[FEATURE] Add setting to disable automatic background task completion notifications

Problem

When a background command completes, Claude Code automatically injects a notification into the conversation:

⏺ Background command "my-command" completed (exit code 0)

Claude then responds to this with something like "Great, that completed as expected" - wasting tokens on a meaningless response.

This is especially annoying when running games, servers, or other long-running processes that the user manually closes. Every single time, Claude responds to the completion notification even though no response is needed.

Real-World Example

1. User asks Claude to run their game for testing
2. User plays the game, then closes it
3. Claude Code injects: Background command "Test game" completed (exit code 0)
4. Claude responds: "The game closed successfully with exit code 0, which is expected..."
5. Tokens wasted, user annoyed

Attempted Workarounds That Don't Work

Hooks/Plugins

I attempted to create a plugin using the hooks system (PostToolUse, Notification events) to intercept and suppress these messages. This doesn't work because the notifications are injected at the system/conversation level, bypassing the hooks pipeline entirely.

Environment Variables

There doesn't appear to be an environment variable to disable just these notifications.

Proposed Solution

Add a setting to disable automatic background task completion notifications:

// In ~/.claude/settings.json or via CLI
{
  "disableBackgroundTaskNotifications": true
}

Or via CLI:

claude config set --global disableBackgroundTaskNotifications true

Behavior When Enabled

• Background tasks still run normally
• Completion notifications are not injected into the conversation
• Users can still check task status manually via /tasks or TaskOutput tool
• Error/failure notifications could optionally still be shown (or have a separate setting)

Related Issues

• #11716 - Background Bash Processes Cause Infinite System-Reminders and Token Exhaustion
• #12813 - Background shell reminders persist for completed shells
• #13249 - Background shell reminders persist after KillShell
• #12302 - Background Bash system-reminders persist after process completion

Why This Matters

• Token cost: Each unnecessary response wastes tokens
• User experience: Interrupts workflow with useless confirmations
• Context pollution: Fills up context window with noise

Environment

• Claude Code version: latest
• OS: macOS
• Shell: zsh