Nonstandard complex numbers and categoricity Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, and by the construction as a quotient of $\mathbb{C}^\mathbb{N}$ we see it's of cardinality $\mathfrak{c}$. The theory of algebraically closed fields of a fixed characteristic is categorical, so this shows ${}^*\mathbb{C}$ is isomorphic to $\mathbb{C}$ in the category of fields.
I'm trying to understand how to interpret this fact in terms of the nonstandardness of ${}^*\mathbb{C}$, namely that $\exists x\in {}^*\mathbb{C} \forall r\in\mathbb{R} x \bar x<r$. 
Question: Am I reading the above correctly to imply that there exists a hyperreal-valued "absolute value" on $\mathbb{C}$ which takes on infinitesimal, standard, and infinite values?
This seems impossible, because the absolute value would have to be infinite, finite, or infinitesimal on real lines in $\mathbb{C}$, and then the triangle inequality would close, for instance, the infinitesimal part under sums. Would we just get a strange decomposition of the plane into three unions of lines, according to which piece of ${}^*\mathbb{R}$ our absolute value fell into? It remains unclear to me that such a decomposition is possible.
 A: All ultrapowers of $\mathbb{C}$ of cardinality the cardinality of the continuum are isomorphic to $\mathbb{C}$ as fields. They are not all isomorphic to $\mathbb{C}$ as fields with additional unary function $\text{Conj}$, the conjugate function.  
We can consider the full structure on $\mathbb{C}$, by adding function symbols, relation symbols for every function and relation on $\mathbb{C}$, including a unary function symbol  $\text{Conj}$. The ultrapower $M$ of $\mathbb{C}$ is an elementary extension of $\mathbb{C}$ with respect to this extended language, and the elements of $M$ that satisfy  $\text{Conj}(x)=x$ are a non-standard model of analysis. But of $\varphi$ is a field isomorphism of $\mathbb{C}$ onto $M$, there is little connection between what $\varphi$ maps $\mathbb{R}$ to and the non-standard model. 
An analogy may be useful. As models of the theory over the empty language, any two structures of the same cardinality are isomorphic. However, such an isomorphism says nothing useful about the relationship between two groups of the same cardinality. 
