Legitimacy of drawing a complex curve like a plane curve

Question

In algebraic geometry, we often consider a complex algebraic curve, and in order to get some intuition, we often draw it on the plane as if it were a plane curve. In most cases it turns out that the plane drawing really helps us thinking of the curve for instance when counting the number of intersections. Then I wonder, to what extent the plane drawing is legitimate (or not misleading) to "represent" the complex algebraic curve?