In Shafarevich's Basic Algebraic Geometry, he gives an example of why we use algebraically closed fields. He considers a circle $C$ and a point $P$ outside of it. He constructs the polar of $P$ as the line joining the points of contact of the two tangents to the circle that pass through $P$. If $P$ is on the circle, the tangent and the polar are the same. But these constructions can be expressed by algebraic relations between the coordinates of $P$ and the equation of $C$, and so are valid when $P$ is inside the circle.
But now, he says, the points of tangency of the tangents have complex coordinates and can't be seen in the figure in his book. But given that the data was real, we can realize certain facts: that the points of tangency must be conjugates and the polar is the line through them, and that the polar can be seen as the locus of points whose polar line passes through $P$.
Now, is there a picture for this situation in $\mathbb{C}$ where there is none in $\mathbb{R}^2$? I feel like I'm missing the point with no geometric intuition present to "refer" to. What are these algebraic relations that define this construction?