  We explore a combinatorial theory of linear dependency in complex space,
"complex matroids", with foundations analogous to those for oriented matroids.
We give multiple equivalent axiomatizations of complex matroids, showing that
this theory captures properties of linear dependency, orthogonality, and
determinants over C in much the same way that oriented matroids capture the
same properties over R. In addition, our complex matroids come with a canonical
circle action analogous to the action of C* on a complex vector space. Our
phirotopes (analogues of determinants) are the same as those studied previously
by Below, Krummeck, and Richter-Gebert and by Delucchi. We further show that
complex matroids cannot have vector axioms analogous to those for oriented
matroids.
