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We prove that for any weighted backward shift B = B w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w n ) satisfies $$ \sup_{n} {\left| {w_{1} w_{2} \ldots w_{n} } \right|} = \infty $$ , the conjugate operator $$ C_{B} :S \mapsto BSB^{*} $$ is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an $$ S \in S(H) $$ such that $$ \{ C^{n}_{B} (S) = B^{n} SB^{*n}\} _{{n \geq 0}} $$ is dense in S(H). We generalize the result to more general conjugate maps $$ S \mapsto TST^{*} $$ , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.