Update: Earlier, I said that the original APK Downloader extension stopped working because it uses Manifest version 1 and that Google Chrome dropped support for all extensions that use Manifest version 1. When I noticed, I checked the author's page for any updates.
The original version stopped working as of Google Chrome version 20+ because starting with this Chrome version, all Google Chrome extension require to use the manifest version 2. It allows you to download APKs from the Play Store webpage via your desktop browser directly onto your computer. This is an updated (and slightly modified) version of redphx's awesome Google Chrome extension APK Downloader.
