Update - the more advanced extension (Spotify Web Extension has remote-control abilities. It uses Google Cloud Messaging to setup a push channel. When you want to remote control another computer, my server wakes up your extension using the channel, and then sets up a relay between the two (a WebSocket connection on both ends).
I just published a very simple extension to skip to the next track in the spotify web player. Here is the chrome web store link. The source code is also on github.
Make sure to set a keyboard shortcut for easiest use (go to the chrome extensions page (type chrome://extensions in the URL Bar) and on the bottom click “Keyboard shortcuts”. Enjoy!
I’m working on a more featurefull extension that will allow you to do more than just skip tracks :-)
