It is well known that there are wild automorphisms of the complex numbers, and that defining the topology on the complex numbers gets rid of all these wild automorphisms. Using this fact, I proved that any nontrivial automorphism of the field of complex numbers does not preserve the set of complex numbers with positive imaginary part, a set closed under addition (By "preserve" a subset I mean that the image of the subset under the automorphism should be the subset itself). I also proved that any nontrivial automorphism of the field of complex numbers does not preserve the set of complex numbers that is a semicircular arc with radius $0.5$ combined with a filled circle of radius $0.25$ centered at the origin, a set closed under multiplication.
My question: is there any such automorphism-killing subset of the complex numbers that is closed under both addition and multiplication?