There are already a number of apps that plug into Android's notification service to mirror the notification shade on your desktop, Pushbullet being perhaps the most prominent example. Notifications+ is a new entrant in this category, but it focuses only on feature-rich notification mirroring.
Notifications+ requires a Chrome extension like both Pushbullet and Krome, but it has a few settings the others don't yet support. You can control the popup timeout on the desktop side on a per-app basis. For example, Gmail could stay up for 30 seconds and Hangouts could disappear after only 5 seconds. Alerts can even be ignored by the Chrome extension if you're not on WiFi, or if the device's screen is on (Krome also does the WiFi thing).
This app also supports dismissing notifications on the device from your computer (as does Pushbullet). Setting up Notifications+ includes a private password that automatically encrypts all your notifications – that's an optional feature in Krome. The basic notification mirroring is free so you can test Notifications+, but the fine-grained settings require a $2.42 pro key. It's not the first app to do this, but it has a nice combination of features worth checking out.