The complex plane does have infinite memory. A Turing machine can perform all discrete computations with a single bit. An idealized analog computer can compute

The complex plane does have infinite memory. A Turing machine can perform all discrete computations with a single bit. An idealized analog computer can compute anything with a "single" complex plane. Our (mis)conception of there being an "infinite memory problem" when the complex plane is infinite is analogous to flatlanders getting hung up about a "discrete bit memory problem."