The theory of algebraically closed fields, ACF, expressed in the language of rings, L(+,×,0,1), is the model completion of the theory of fields. As such, the theory of algebraically closed fields serves as a foundation for the model-theoretic study of algebraic geometry.
The completions of ACF, ACFp for p prime or zero, are given by specifying the characteristic.
The theory ACFp is categorical in each uncountable cardinal. The proof of this theorem generalizes to any strongly minimal theory immediately once the relevant notions of algebraic independence, dimension, etc. are expressed strictly model theoretically.