Ionic compounds are those where ions interacted with each other, through electron exchange, and bonded together. In other words, ionic compounds have ionic bonds. These compounds are conductors of electricity, when their ions are free to move. Thus, they do not conduct electricity when they are solid. However, when they are dissolved in water or a solution, they break down into ions and conduct electricity. Similarly, in the molten state, the ions are free to move and hence can conduct electricity. A good example is common salt, sodium chloride (NaCl). In its powdered form, it does not conduct electricity. However, when we dissolve it in water, sodium and chloride ions are dissociated and conduct electricity easily.
Thus, ionic compounds are capable of conducting electricity in molten form. Note that these compounds have very high melting and boiling points.
Hope this helps.