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As I understand, an affine plane is the set of all 2-tuples of elements of some algebraically closed field $k$. Can you visualize this plane with coordinates, like the euclidean plane $\mathbb{R}^2$? In particular, can you visualize curves? For example, can we always think of the zero set of $x^2+y^2-1$ as a circle, even though $k$ is not $\mathbb{R}$?

When I consider $k=\mathbb{C}$, it seems like the "Affine plane" would be $\mathbb{R}^4$ in this case, so I don't think it looks like a plane at all...

Sometimes I see "affine plane curves" depicted on a plane which is labelled exactly like $\mathbb{R}^2$, so this is adding to my confusion.

user138022
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$\Bbb R^2$ is the most readily available plane for visualization purposes. So it's a convenient place to draw things, even though it's not technically correct. Often intuition can be gleaned from it anyways, although one always has to be aware that it's only a cross section of $\Bbb C^2$, and not even close to correct for other $k$.

I only rarely see attempts at faithfully drawing any affine planes over algebraically closed fields. And they tend to look pretty abstract, and not much like the Euclidean plane we're used to at all.

Arthur
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