Is the Cx4 .45 capable of firing .45 Super ammunition, or if not, is a conversion available?

No. Due to significant variations between .45 ACP and .45 Super ammunition, .45 Super ammunition should never be used in the Cx4 carbine. Beretta does not recommend any type of adapters as it would require modifications to the barrel and chamber as well as the firing pin. Modifications to your Beretta firearm will void the factory warranty.