The paper develops, within a new representation of Clifford algebras in terms of tensor products of quaternions called hyperquaternions, several applications. The first application is a quaternion 2D representation in contradistinction to the frequently used 3D one. The second one is a new representation of the conformal group in (1+2) space (signature +-- + − − ) within the Dirac algebra C_{5}\left(2,3\right) \simeq \mathbb{C\otimes H\otimes H} C 5 ( 2 , 3 ) ≃ ℂ ⊗ ℍ ⊗ ℍ subalgebra of \mathbb{H\otimes H\otimes H} ℍ ⊗ ℍ ⊗ ℍ . A numerical example and a canonical decomposition into simple planes are given. The third application is a classification of all hyperquaternion algebras into four types, providing the general formulas of the signatures and relating them to the symmetry groups of physics.