[math-fun] complex #'s explore circle group; quaternions explore SU(2) ?

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 ???