Can you install laminate over rough slate?

I would definitely recommend against this. While laminate can be installed on top of tile, it requires that the floor is level and even. So, if your slate is rough and uneven, it is not a good subfloor to install any hard surface. If you install the laminate directly on top of an uneven floor like this, it will most likely crack and move a lot when you are walking on it.

However, an alternative (albeit an expensive one), is to add self leveling cement on top of the slate and level out the floor, and then you could install the laminate on top. This might be less expensive and less difficult vs. trying to remove the slate, but it is not a cheap option.