Improve quality of ComplexPlot contours

Since this option is not in the documentation, @J.M.'s comment should be in an answer:

Try cranking up RasterSize, e.g. RasterSize -> 900. – J. M.'s ennui♦ 14 hours ago

ComplexPlot[1/(x^2 + 1), {x, -2 - 2 I, 2 + 2 I}, 
 ColorFunction -> {ColorData["RedBlueTones"], "CyclicLogAbsArg"}, 
 RasterSize -> 900]

Crank it up more, if dissatisfied.

One might discover the option by pondering the undocumented things in:

ComplexPlot // Options
(*  {..., RasterSize -> Automatic,...}  *)

It's a long list, though.

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Note by J.M.:

The PlotPoints/MaxRecursion settings do not affect the resolution of the domain coloring plot here. Instead, they only affect the resolution of the boundary (as rendered with BoundaryStyle) and mesh (as rendered with Mesh/MeshFunctions/MeshStyle).

RasterSize is independent of these, and controls the fineness of the sampling of the domain coloring image itself. It could certainly be argued that it should either be documented, or should at least have a relation with the more natural PlotPoints option.