How do I sperate an object while having both objects retain the same face? When I pres p and separate the cube I want the face withe raise pyramid to be on the bottom part of the cube and also on the top.  I want the parts of the cube to fit back together.

I hope my question is clear enough to understand as I wasn't exactly sure how to phrase it.
Select the faces you want, then duplicate them with Shift+D. Offset in the z-direction if you want them to stay lined up (Shift+D, Z, move mouse, left-click to make it stick). Once you've done that, you'll have something that looks like this:



From here you can separate the selected faces from the object, then go down to the original and delete the faces you don't want. (Another option is to duplicate the object and delete the unwanted faces in each.)