In his classification of complete minimal surfaces of finite total curvature -8πi from 1992, Francisco López constructs also 2-ended spheres whose ends have winding number 2.
The simplest case is trivial: You can just take the double cover of a catenoid, which will look exactly like the catenoid. He finds, however, a 2-parameter family of such surfaces, the most symmetric of which has Enneper-Weierstraß data
F. López: The Classification of Complete Minimal Surfaces with Total Curvature Greater than -12π, Trans. Amer. Math. Soc. 334, 49-73, 1992