  For an algebraic K3 surface with complex multiplication (CM), algebraic
fibres of the associated twistor space away from the equator are again of CM
type. In this paper, we show that algebraic fibres corresponding to points at
the same altitude of the twistor base $S^2\simeq \mathbb{P}^1_\mathbb{C}$ share
the same CM endomorphism field. Moreover, we determine all the admissible
Picard numbers of the twistor fibres.
