Continue game on another PS4

What you are looking for is PS4 to PS4 remote play and this is not possible. There is a solution though. While it didn't get much love, the PlayStation TV will allow you to do exactly what you want to do.

The PlayStation TV is pretty a PS Vita but it plugs into your TV. It supports remote play just like the vita does. This means you can pause your game on the main PS4 and move over to your other room where PS TV is setup, trigger remote play and you are good to go.

The PlayStation/Vita TV costs like $50 so it's a cheaper alternative to owning 2 PS4s.

There really aren't many choices on this. The best way to actually to do what you are wanting to accomplish is via usb/cloud transfer files between PlayStations. The only way to do it without transferring saves is to have one PlayStation hooked up to two different tv's (which isn't desirable unless you have really long cords on hand!)