who is he
It definitely would get far too complicated. Immunities would become overpowered, STAB would be even more lethal, and Pokemon would need about five times as much HP as they already have.
We already have attacks that take into consideration Sp. Attack and Defense, so maybe some future attacks will have some unique stuff going on with their typing (and, hey, we already have attacks like Scald and Hidden Power), but not to the extent of dual-type attacks.