What compound conducts electricity only in the molten state?
Salts or oxides, if they're insoluable in water.
If soluable, they would also conduct electricity as a solution, because the ions can move freely.
Molten salts and oxides also have free-moving ions, so they will also conduct electricity.
Both will be electrolysed, as the ions lose their charge at the electrodes. This is the way aluminium is made out of molten aluminium oxide. The oxygen produced at the other (graphite) electrode will react with it to produce carbon dioxide.