These examples are from Hermann Karcher’s Tokyo Notes, mildly generalized. We have a sphere with two Enneper ends. This leaves plenty of freedom so that the ends can be rotated against each other:
By choosing the angle wisely, the surfaces become a little bit less not embedded, if this makes sense. The same works of course with higher order Enneper ends.
This twist shows that we here have a non-contractible moduli space of minimal surfaces, something that hasn’t been observed for embedded surfaces.
All the surfaces here still have horizontal lines as normal symmetry lines and come besides the twist parameter with a parameter that stretches the catenoidal neck between the Enneper ends.