Of course. North is north and south is south wherever you are on the planet. HOWEVER... there is such a thing as magnetic field DIP. Your compass needle aligns itself to be parallel to magnetic field lines. But if you recall from grade school study of magnetism, remember how those iron filings line up on the bar magnet. Near the poles (north or south), the field lines are nearly vertical. Near the mid point (equator), the lines are nearly parallel to the surface. There is
information on this on this web page.
Your compass neeedle behaves the same way. It must be slightly weighted on one end to keep it level, to compensate for the magnetic dip. If not, the needle will want to align itself with the magnetic field, which could cause it to strike the bottom of the case, depending on where you are on the planet. The balance weight is the opposite in opposite hemispheres. When you buy a compass in North America, it may not work very well in Patagonia because the needle balance is wrong. You have to either buy a compass balanced for dip in South America, or you can now buy what is known as a universal or global compass such as the
Suunto MC-2G (G for Global) , which is supposed to work everywhere by the mechanism built in.