We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields K and characterize the "badly approximable" numbers, (z such that there is a C = C(z) > 0 with |z − p/q| ≥ C/|q| 2 for all p/q ∈ K), by boundedness of the partial quotients in the continued fraction expansion of z. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in C whose points are badly approximable over K, with effective constants.By other methods (the Dani correspondence), we prove the existence of circles of badly approximable numbers over any imaginary quadratic field. Among these badly approximable numbers are algebraic numbers of every even degree over Q, which we characterize. All of the examples we consider are associated with cocompact Fuchsian subgroups of the Bianchi groups SL 2 (O), where O is the ring of integers in an imaginary quadratic field.

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We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are "reflective" versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. Quadruples and the Super-Apollonian Group2.1. Simple continued fractions. We begin with a brief overview of simple continued fractions on the real line, described from the perspective and in the language we plan to use for complex continued fractions in the remainder of this paper. The purpose is to provide an explicit analogy for much of what follows.Over the integers, the Euclidean algorithm is the iteration of the division algorithm, which, for a, b ∈ Z, returns q, r ∈ Z so that a = bq + r, 0 ≤ r < |b|. Replacing the pair (a, b) with (b, r), we

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