The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as O(p+1,q+1) isomorphic to the nD conformal group with n=p+q. The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory Response to Reviewers: All remarks have been taken into account.