The purpose of the paper is to construct a new representation of dual quaternions called bi-periodic dual Fibonacci quaternions. These quaternions are originated as a generalization of the known quaternions in literature such as dual Fibonacci quaternions, dual Pell quaternions and dual $k$-Fibonacci quaternions. Moreover, some of them have not been introduced until now. Finally, we calculate the generating function, Binet formula and Catalan's identity of the bi-periodic dual Fibonacci quaternions.