Square conformal models for 2 dimensional Euclidean geometry?

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The Poincare disk model is one example of a conformal disk model of 2 dimensional Hyperbolic geometry. I'm wondering if there are conformal square models for 2 dimensional Euclidean geometry. What are some common ones?

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If you just want a local model of Euclidean geometry, this is completely trivial: just take the usual Euclidean geometry of the plane, restricted to a square.

If you want a global model, that is not possible, since a square is not conformally equivalent to the Euclidean plane. (Instead, it is conformally equivalent to the hyperbolic plane. Indeed, every simply connected open proper subset of the Euclidean plane is conformally equivalent to the hyperbolic plane, by the Riemann mapping theorem.)