Hamiltonian dynamics play a key role in the foundation of modern physics and mathematics with wider applications in multiple advanced sciences and technologies.
This paper proposes a conjugate transformation structure and its measurement operators on a hierarchy of multiple levels to support intermediate transforming structures from pairs of logic states as micro-ensembles to feature vector transformations as global measurements.
Using logic equations and pairs of partitions on phase spaces, conjugate 0-1 vectors provide hypercomplex number systems. Multiple operators can be created and linked with Hamiltonian operators.
The main constructions of conjugate transformation structures are described and complex conjugate operators are discussed under a pair of symmetric and antisymmetric parameters with O(2^{2^n}x2^N); 1 =< n =< 2m structural complexity.
Using new operators, the Yang-Mills equations are briefly described as an example.
