Construction of maps $f:S^3 \to S^2$ with arbitrary Hopf invariant?
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
$$x_1+x_{2}i+x_3j+x_4k$$
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!