The unit quaternions, {x + yi + zj + wk | x^2 + y^2 + z^2 + w^2 = 1}, form the unit sphere S^3 in R^4, a 3-(real)-dimensional manifold. As an odd-dimensional manifold, S^3 is not likely to have a convenient parametrization by complex numbers, exactly, but as SU(2) of course we can write (a -b*) SU(2) = { ( ) in M_(2x2)(C)| |a|^2 + |b|^2 = 1} (b a*) But you probably knew that. I'm not sure of the exact meaning of "enumerated uniformly" here. Or what enumerated uniformly by a random walk means. But there is no differentiable map from a real 2-dimensional manifold, or from a complex 1-dimensional manifold, to S^3 that is *surjective*. (Proof: The non-critical points are carried to S^3 by a locally one-to-one map; the critical points are carried to a set of measure zero by Sard's Theorem.) The group SU(2) is the double cover of the orientation-presrving rotation group SO(3) of R^3. One can think of any element of SO(3) as defined by an ordered pair (a, theta) where a is the axis of counterclockwise rotation, and theta is an angle in the range (0, pi). Such a pair can be represented by the ball B = {(x,y,z) in R^3 | x^2 + y^2 + z^2 <= pi} where all points with theta = 0 have been identified to the origin. We still have each rotation by angle pi represented in B as a pair of antipodal points. So to get SO(3) we need to identify these pairs each to a single point. ----- There is a related question of finding a DENSE curve in some space: For any manifold M one may ask for a smooth (C^oo) curve that is dense in M. It's interesting to also require that the curve be *uniformly distrbuted* in M. Not a lot is known about that. But I do believe the probabilist Persi Diaconis has a number of theorems on the subject of how a random walk on a Lie group becomes dense. —Dan Henry Baker wrote: ----- We know that the circle group is enumerated uniformly by cos(t)+i*sin(t), for i in [0,2*pi) using a single (real) parameter t. Is there a way of uniformly enumerating the unit quaternions * with 1 parameter (probably some sort of random walk) ? * with 2 parameters (probably some sort of random walk) ? * with 1 complex parameter ????? * with 3 parameters ??? -----