I saw a picture of a small object near the edge of a circle along with its circle inversion, and it looked a lot like a reflection.
That’s when I remembered that they’re both anti conformal involutions and realized that inversions could be thought of as “reflecting about circles”.
Are there analogous anticonformal involutions for other conics (and maybe more general shapes)?
If you're looking for anticonformal "reflections" across curves, they exist for analytic curves and are called Schwarz functions. The cited article mentions the reflections for a circle and ellipse, and the ellipse case is worked out in Needham's Visual Complex Analysis. See also the Schwarz reflection principle,
An animation of this (built in Geogebra) is shown in this tweet. You'll notice that the "reflection" works quite nicely near the curve, but behaves somewhat differently farther away. I've played around with this and implemented Schwarz reflection in other curves such as $x^4+y^4=1$.