I love pure math but detest teaching. What non-academic options do I have after finishing a PhD in Complex Algebraic Geometry?
It might be too early for you to claim you detest teaching and believe this could remain for the rest of your life. Teaching  does not only consist in "repeating and repeating again", and after a while you can find interest in teaching. Especially if you try novel ideas with your best students. 

Part-time teaching can be an interesting way to stick to your field of interest, while doing something else. 

Then,  it is unlikely that you get paid for a very focused topic outside academia. Meanwhile, there exists a belief, summarized by E. Wigner in:


  The unreasonable effectiveness of mathematics in the natural sciences


Many people outside academia have some fear of mathematics. Yet, it has found many applications. And those people need some fellow that master black magic. Topology has found applications in data science, algebraic geometry has potentials in cryptography or computer, and possibly other domains:


What are some applications outside of mathematics for algebraic geometry?
“Real”-life applications of algebraic geometry


So why not invest time in applied sides, while keeping connexity to mathematics, at least closer that with dealing with white powder (I mean flour for bread of course)?

Finally, in some countries (like France), there exist research  institutions where teaching is not mandatory (CNRS, INRIA). You could apply there, say for a post-doc position, and see what happens.