What is the name of this ionic compound Cu (MnO4)2?

Cu is copper
Mg is magnesium
O4 is oxygen

Copper and magnesium oxide?

Nope, MnO4 is the managanate (VII) ion, hence, this becomes copper manganate (VII)

It's Copper (ii) Permanganate
Copper (ii) is Cu with a charge of 2+
Permanganate is MnO4 with a charge of 1- (which means that you need two of them to balance out making it (MnO4)2 and not just MnO4)