  In this conceptual paper, we discuss quantum formalisms which do not use the
famous Axiom of Choice. We also consider the fundamental problem which
addresses the (in)correctness of having the complex numbers as the base field
for Hilbert spaces in the K{\o}benhavn interpretation of quantum theory, and
propose a new approach to this problem (based on the Lefschetz principle).
Rather than a Theorem--Proof--paper, this paper describes two new research
programs on the foundational level, and focuses on fundamental open questions
in these programs which come along the way.
