I just read through Chapter 1 of that book, and none of those terms have anything to do with complex analysis in context. This is a case of two different fields using the same word to mean different things.
The word "singularity" is used whenever some distribution (charge, electric field, etc.) becomes infinite at a point. Though singularities do appear in complex analysis, that is by no means the only place they appear. In the particular case of classical electrodynamics, the impact of singularities is described by the use of Gauss's Theorem, which is derived from the divergence theorem of vector calculus, not from complex analysis. The existence of singularities does not by any means imply the use of complex analysis.
I did not find a single usage of the word "pole" outside of the phrase "$2^n$-pole", which I will address next. In general, though, "pole" in electromagnetism doesn't necessarily refer to a singularity in a complex function like it does in complex analysis. Oftentimes it is used to specify the location of maximum field strength on the surface of a magnet (the "north and south poles" of a magnet, in common usage).
The phrase "$2^n$-pole" is derived from the multipole expansion of a potential. This expansion is used when the small-scale angular features of a potential are irrelevant, and can often massively simplify the study of a complicated charge distribution. The constants in this expansion are called the "monopole moment," "dipole moment," "quadrupole moment," "octopole moment", and so on, in general being referred to as $2^n$-pole moments, for the $n$th term in this expansion. The monopole moment measures how much the distribution looks like a single charge, while the dipole moment measures how much it looks like a pair of opposite charges; the quadrupole moment measures how much the distribution looks like a square of alternating charges, and so on, describing progressively finer angular features.
This, again, has very little to do with complex analysis. Though you can technically call the multipole expansion a Laurent series, the domain of the function is restricted to real values of $r$, so none of the properties of the Laurent series are actually used, making it basically just a Taylor series after a substitution. You don't need to know anything about complex analysis to use them.
That said, it's still a good idea to learn about complex analysis, as it's quite useful in the more advanced areas of physics. Just be careful not to assume that a particular term means the same thing in every field.