You can show this with ring magnets. Put them on a stick, and they do indeed float over each other. However, the stick is required, that's your extra axis of stabilization--remove it, and the magnets simply flip over and jump to each other, no longer floating. The phenomenon of diamagnetism provides a simple, inexpensive solution to this problem and gives fairly dramatic results.
Most people are familiar with ferromagnetic materials--iron, steel and permanent magnets. These materials are attracted by magnetic fields, and can keep their magnetism after exposure to the field. Diamagnetic materials include carbon-graphite, water, protein, DNA, wood, bismuth, silver, diamond and gold. When exposed to a magnetic field, these materials induce a weak magnetic field in the opposite direction. Superconductors are perfectly diamagnetic, but require liquid nitrogen to work--not something that most folks have a jug of in their fridge!
Diamagnetism is the key to this experiment, and provides the extra stabilizing force needed. We'll be using carbon-graphite and bismuth because they have the strongest diamagnetic effect.