The complexity of isomorphisms for computable and decidable structures plays animportant role in computable model theory. Goncharov  defined the degree ofdecidable categoricity of a decidable model (M to be the least Turing degree, if it exists, which iscapable of computing isomorphisms between arbitrary decidable copies of (M. If this degree is 0, we say that the structure (M is decidablycategorical. Goncharov established that every computably enumerable degree is thedegree of categoricity of a prime model, and Bazhenov showed that there is a prime model with nodegree of categoricity. Here we investigate the degrees of categoricity of various prime models withadded constants, also called almost prime models. Werelate the degree of decidable categoricity of an almost prime model (M to the Turing degree of the set C(M) of complete formulas. We also investigate uniformdecidable categoricity, characterizing it by primality of (M and Turing reducibility of C(M) to the theory of (M.