Let \(M^{4n}\) be a \(4n\)-dimensional manifold. An almost hypercomplex Hermitian structure on \(M^{4n}\) is defined by a pair \((H,g)\) on \(M^{4n}\), where \(H=(J_\alpha )_{\alpha =1,2,3}\) is an almost hypercomplex structure on \(M^{4n}\) and \(g\) is a Riemannian metric on \(M^{4n}\) which is Hermitian with respect to \(H\). An almost quaternionic Hermitian structure on \(M^{4n}\) is defined by a subbundle \(Q\) of rank \(3\) in the vector bundle of the tensors of type \((1,1)\) on \(M^{4n}\), such that \(Q\) is locally generated by almost hypercomplex structures \(H\). Moreover, there exists a Riemannian metric on \(M^{4n}\) which is Hermitian with respect to any generating almost complex structure \(H\). A quaternionic-like structure on \(M^{4n}\) is one of the following structures: the almost hypercomplex structure with the structural group \(GL_n({\mathbb{H}})\), the almost quaternionic structure with the structural group \(Sp_1.GL_n({\mathbb{H}})\), the almost hypercomplex unimodular structure with the structural group \(SL_n({\mathbb{H}})\), the almost quaternionic unimodular structure with the structural group \(Sp_1.SL_n({\mathbb{H}})\), the almost hypercomplex Hermitian structure with the structural group \(Sp_n\), the almost quaternionic Hermitian structure with the structural group \(Sp_1.Sp_n\). The author studies the properties of some canonical \({\mathcal D}\)-connections or families of \({\mathcal D}\)-connections for the quaternionic-like structures. Further he presents some extensions of the definitions and results to the case of conformal almost quaternionic Hermitian structures and to the case of Lichnerowicz type connections.