Searching for Arduino and PS4 on google, I find always the same construction using a USB host shield and a Bluetooth dongle.
But I'd like to connect a PS4 Controller to a HC-05 device. Is it possible? If not, why not?
The HC-05 uses the Bluetooth Serial Port profile (SPP). The PS4 controller doesn't.
Hence you need something that is capable of using the same profile that the PS4 controller uses (HID?). And to that end you need something a little more configurable than a simple HC-05. One option is, as most people use, a USB Host shield with a Bluetooth dongle that can then be configured to communicate using the right profile.