Complex numbers and Nonstandard Analysis

The complex numbers can be defined as numbers of the form $x + i y$ where $x$ and $y$ are real. All constructions of standard analysis work in non-standard analysis, so this statement remains true in the non-standard model: hypercomplex numbers are numbers of the form $x + i y$ for hyperreal $x$ and $y$.

Whether it's useful to single out those complexes with standard real and infinitesimal imaginary part, I think it unlikely. Finite real and infinitesimal imaginary part is more likely to be useful.

For example, in complex analysis, one often constructs contours that include paths of the sort $x + i \epsilon$ to approximate the real line, or circular arcs $a + \epsilon e^{i \theta}$ to approximate a point, where $\epsilon$ is a small positive real number, and then takes the limit as $\epsilon \to 0$.

Replacing $\epsilon$ with a real infinitesimal would probably be useful.