Use a floating window to watch for incoming email messages in Mail even if it isn’t running. You can delete messages, mark them as junk or reply to them. Pick which accounts you want to monitor, and choose a theme for the window’s appearance.
Email app promises to keep you inbox organized with categorized messages, filters that only let through the important notifications, and powerful search. Also available for iOS.
From the makers a Firefox, an open source email client that features tabbed browsing, filtering and timeline tools, plenty of customization, and robust security measures.