Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields?
Everest and Ward give examples of points of height 0.01032... and 0.009721... on curves over Q(w) for w a cube root of unity or the golden ratio respectively. I have made a modest improvement in the latter case, recovering a point of height 0.009128... .
In the context of the elliptic Lehmer problem the aim is to minimise dh(P) for d the degree of the number field, so working over quadratic extensions a point would have to have height less than 0.005 to be competitive with the examples in Elkies' table. Are there any examples?