The author mentioned that the theory of the complex field is categorical, but I didn't see them directly mention that the theory of the real field isn't - for every cardinal there are many models of the real field of that size. My own, far less qualified, interpretation, is that even if the complex field is just a convenient tool for organizing information, for algebraic purposes it is as safe an abstraction as we could really hope for - and actually much more so than the real field.
The real field is categorically characterized (in second-order logic) as the unique complete ordered field, proved by Huntington in 1903. The complex field is categorically characterized as the unique algebraic closure of the real field, and also as the unique algebraically closed field of characteristic 0 and size continuum. I believe that you are speaking of the model-theoretic first-order notion of categoricity-in-a-cardinal, which is different than the categoricity remarks made in the essay.
I believe the author does talk about the first-order model theoretic perspective at one point, but yes, I was referring to that notion.