In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent representation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames. Though Clifford algebras are all related to quaternions via the Clifford theorem, a biquaternion formulation of differential geometry does not seem to have been formulated so far. The paper develops, in 3D Euclidean space, a biquaternion calculus, having an associative exterior product, and applies it to differential geometry. The formalism being new, the approach is intended to be pedagogical. Since the methods of Clifford algebras are similar in other dimensions, it is hoped that the paper might open new perspectives for a 4D hyperbolic differential geometry. All the calculi presented here can easily be implemented algebraically on Mathematica and numerically on Matlab. Examples, matrix representations, and a Mathematica worksheet are provided.