The wellknown Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$ has explicit constructions involving the geometry of $C^2$ and intersections of complex lines with the $3$sphere. They don't seem to generalize easily to "higher" Hopf maps from $S^3 \rightarrow S^2$ with Hopf invariant not equal to one. Are there any simple expressions for those maps?
You can get them by precomposing with a degree n map from S^3 to itself. In particular, this gives an interpretation in terms of the group structure: if h:S^3 \to S^2 is the Hopf map (which is just modding out by the subgroup S^1=U(1) of S^3=Sp(1)), then a map of Hopf invariant n is given by x \mapsto h(x^n), where x^n is using the group multiplication on S^3. 


Actually, yes, there is a construction involving complex projective line. Consider all points (x_{1}, x_{2}, x_{3}, x_{4}) on a 3sphere in the 4dimensional space. Our goal is to map them to To do this, take a quaternion
raise it to the 

