It actually is possible without any adapter by just installing some files on your SD card.
The source files and some support information is here.
https://github.com/cathery/sys-con
BUT, I must stress that this only allows WIRED controllers via USB. Wireless doesn't work YET. Only USB is supported.
So you CAN use a Dualshock 4, as long as you have the charge cable.
It's probably best to buy a dongle for a much simpler solution as Dorian stated.
I use a Mayflash Magic NS myself and it's great. I only wish it supported more than one DS4 simultaneous per dongle.