The well-known 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 (x1, x2, x3, x4) on a 3-sphere in the 4-dimensional space. Our goal is to map them to $S^2$ which is the same as $CP^1$
To do this, take a quaternion
raise it to the $n$-th power (this is that group law on a 3-sphere) and decompose back into two complex numbers $z_1+z_2j$ . Now $z_i:z_i$ is a point of a complex projective line, that's it!