We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results of the authors on the de Rham and elasticity Hilbert complexes with mixed boundary conditions and results of Pauly and Zulehner on the biharmonic Hilbert complex with empty or full boundary conditions.