That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.

In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)