Hello,
Depends on what type of structure you have and what level of seismic forces you have in the column... But in the most situations yes the foundation need to be designed at the capacity of the column so the column can rotate and achive it's purpose in disipation of seismic energy. In all the process from yielding to breaking the foundation have to sustain the capacity of the element.
In some cases where you have the overstrenght larger then the R factor of seismic reduction you will not have plastic hinge and your column will stay in elastic domain, there no, you dont have to calculate the foundaiton with the forces from capacity of the column you will use elastic forces from earthquake.

See AISC 341-10 D2.6c (especially the commentary). Foundations should be designed to remain elastic unless they are detailed for ductility per ACI 318-11 Chapter 21.

Sort of. In seismic design, a plastic hinge in the column is bad. The hinge should form in the beams. But, the baseplates and there attachment should be designed for the plastic moment capacity of the frame (or amplified seismic loads - overstrength). The foundations (and supporting soil), for California Hospital's need to be designed for the amplified loads. But, I don't think that most buildings are required to have the supporting soil resist the amplified loads.

Thanks for your input

In seismic design you are always assuming a plastic hinge forms at the base of a column. The strong column weak beam criteria is to prevent a single story mechanism vs. forming plastic hinges in the beams and all levels and at the base. A comparison between overstrength and the plastic capacity of the column you be check and the lower of the two should be selected.