The problem of categoricity of AEC formalizing analytic covers of complex algebraic varieties will be discussed. The recent proof of the categoricity of pseudo-analytic covers of Shimura curves in a natural setting (joint with Chris Daw) contributed to much clearer and more positive view at the following three issues: What does a pseudo-analytic categoricity theorem mean in terms of algebraic and analytic geometry? What is and how far the general pseudo-analytic categoricity conjecture goes? Is there a good model theory in the study of pseudo-analytic structures from this point of view?