Here's an explicit construction method, though I'm not sure if it's truly that useful for visualization.
Designate the north pole of the unit 2-sphere $\Bbb S^2\subset\Bbb R^3$ as the vector $n$. We can identify the complex plane with the sphere minus a single point, ie $v:\Bbb C\cong \Bbb S^2\backslash\{n\}=S$. Then we seek a family of maps indexed by $v\in S$ of the form $l_v:\Bbb C \xrightarrow{\sim} v^\perp$, where by $v^\perp$ we mean the orthogonal complement of the subspace generated by $v$ (in other words, the plane containing the origin perpendicular to $v$).
We can send $(z,w)\in\Bbb C^2$ to the line with direction $v(z)$ and displacement from the origin (normal to the direction $v$) given by $l_v(w)$. The line can be described as the coset $\Bbb R v(z)+l_{v(z)}(w)$ within $\Bbb R^3$.
It's hard to think of a "canonical" family of maps $l_v$ continuous in $v\in S$ though. One way is to use a Gram-Schmidt process to create a dynamically moving frame; define the vector functions
$\begin{cases} r:= \frac{n\times v}{\|n\times v\|}, \\[4pt] s:=\frac{v\times r}{\|v\times r\|}. \end{cases}$
Note that $r,s:\Bbb C\to S$ are both well-defined, are orthogonal to each other as well as to $v$, due to the basic geometric properties of the cross product. We can then write $l_v(a+ib)=ar+bs$. We then send complex functions $f$ to sets of lines in $\Bbb R^3$ by applying our map to $\{(z,f(z)):z\in\Bbb C\}\subset \Bbb C^2$.
Again, I feel the practical utility of this, relative to modern methods of colored surfaces or density plots, is dubious. Perhaps someone else can think of a more natural construction method, one that is easier to visualize and conceptually process, but this is the only thing that comes to my mind.