In a strange turn of events that makes you wonder why it hasn't happened before now, Valve has turned on the Bluetooth modules in its Steam Controllers, so if you ever wanted to play with the Valve gamepad without the included dongle on something other than a Steam Link, now you can.
The whole purpose of the added functionality is to make it so that the Steam Controller can pair up with smartphones that are running the upcoming Steam Link app, whereby gamers can stream their PC games straight to their smartphone. That's a great addition since most PC games are not designed with a touch interface in mind. A controller, even one with a touchpad like the Steam Controller, isn't a true replacement for a mouse and keyboard either, but it's certainly better than the alternative.
If you want to have a play with the new feature yourself, you will need to update the Steam Controller and Steam Client on your PC. To do so, plug in the controller, opt in for the beta-client version of Steam and then when prompted choose to "update firmware now," for the gamepad. Once complete, you'll have a functioning Bluetooth controller.
Once done, you'll need to re-pair your controller with the dongle if you make use of that, but the new feature will unlock the ability to swap between the two wireless modes that the Steam Controller (then) supports. There's the stadard wireless mode, and new Bluetooth Low Energy mode. Switching between them is simple, as PCGamesN points out: you just press A and the Steam button in the center of the device to switch to Wireless, and B and the Steam button to switch to Bluetooth.