  An orthogonal complex structure on a domain in R^4 is a complex structure
which is integrable and is compatible with the Euclidean metric. This gives
rise to a first order system of partial differential equations which is
conformally invariant. We prove two Liouville-type uniqueness theorems for
solutions of this system, and use these to give an alternative proof of the
classification of compact locally conformally flat Hermitian surfaces first
proved by Pontecorvo. We also give a classification of non-degenerate quadrics
in CP^3 under the action of the conformal group. Using this classification, we
show that generic quadrics give rise to orthogonal complex structures defined
on the complement of unknotted solid tori which are smoothly embedded in R^4.
