Saying quaternions require thinking in 4 dimensions seems like a lie with no proof. The geometric product is just the quaternion product broken up into scaler and vector parts.
Quaternions are a concept specific to the 3-dimensional (Euclidean) space, in the same way as "complex" numbers (for whom "binions" would be a more appropriate name) are a concept specific to the 2-dimensional (Euclidean) space.
Neither quaternions nor "complex" numbers have anything to do with a 4-dimensional space of vectors.
Quaternions are a field that is a subset of the 2^3 = 8-dimensional geometric algebra associated with a 3-dimensional space of vectors, while the "complex" numbers are a field that is a subset of the 2^2 = 4-dimensional geometric algebra associated with a 2-dimensional space of vectors.
While vectors are associated to transformations of the corresponding affine space that are translations, quaternions/complex numbers are associated to transformations of the space that are rotations or similarities.
Saying quaternions require thinking in 4 dimensions seems like a lie with no proof. The geometric product is just the quaternion product broken up into scaler and vector parts.
You are right.
Quaternions are a concept specific to the 3-dimensional (Euclidean) space, in the same way as "complex" numbers (for whom "binions" would be a more appropriate name) are a concept specific to the 2-dimensional (Euclidean) space.
Neither quaternions nor "complex" numbers have anything to do with a 4-dimensional space of vectors.
Quaternions are a field that is a subset of the 2^3 = 8-dimensional geometric algebra associated with a 3-dimensional space of vectors, while the "complex" numbers are a field that is a subset of the 2^2 = 4-dimensional geometric algebra associated with a 2-dimensional space of vectors.
While vectors are associated to transformations of the corresponding affine space that are translations, quaternions/complex numbers are associated to transformations of the space that are rotations or similarities.