It's hard to overstate my love for AirDroid, the app that hosts a tiny, web-based interface for managing your phone with a desktop browser. And it just keeps getting better: the latest update has added notification mirroring, a la PushBullet and Notifications+. That means that once you enable the setting in AirDroid and Android, you'll get visual indications of incoming notifications on your AirDroid desktop.
Enabling mirroring is easy - you'll get a prompt the first time you start the updated application, so all you have to do is check the box in the linked settings dialog. At the moment the service has some limits on its functionality, mostly because the alerts appear and disappear in a few seconds, and there's no way to scroll back through them - you'll just have to pick up your phone and check. There's also no audio alert, so if you don't have the AirDroid tab open, you won't get the notification on your computer. At the moment only notifications from Gmail, AirDroid, Hangouts, Google Play Music, and the Android system itself are supported.
Update: It looks like the option to show or hide notification types won't show up in the AirDroid settings menu until a new notification from that app appears while AirDroid is running. According to the developer, AirDroid can show all Android notifications on a desktop browser. Sorry for the confusion.
The latest update also adds a handy usability feature: you can now drag and drop files from Windows or OS X directly onto folders in the AirDroid file browser for an instant upload, so you don't need to bother with the dedicated upload tool anymore. Once you drop the file, it will instantly start the upload process. If you've got fast WiFi or wireless broadband, small files should go over the network quickly enough.
AirDroid is a free download for Android 2.2 or higher. It's free to use over WiFi, but accessing more than 100MB of data over the mobile network per month requires a premium subscription.