Trying to visualize supersingular elliptic curves over GF(p^2)

I'm working on a project in which I'd like to visualize points on supersingular elliptic curves over GF(p^2). I've got a plan for handling the handful of SSECs that are defined on Fp (scatterplot on a torus), but the GF(p^2) ones are stumping me.

My thought is to represent GF(p^2) by affixing sqrt(r) for some QNR r... so having a+br for a, b in Fp, and then somehow representing a map Fp x Fp -> Fp x Fp this way. Since these maps are not very nice & are discrete, I'm not sure how to proceed.