at.algebraic topology - Does a "Chern character" exist for any generalized cohomology theory?

For any (connective) spectrum $E$ one may rationalise it to get a rational spectrum $E_mathbb{Q}$, and a map $E to E_mathbb{Q}$. Now rational spectra split as wedges of Eilenberg-Mac Lane spectra, so one may choose an isomorphism $E_mathbb{Q}^*(X) simeq H^*(X;pi_{-*}(E)otimes mathbb{Q})$, and the rationalisation gives a map
$$ch_E : E^*(X) longrightarrow H^*(X;pi_{-*}(E)otimes mathbb{Q}).$$

For complex K-theory this gives the Chern character, and for real K-theory it gives the Pontrjagin character.

Of course, if $E$ is a ring spectrum so is $E_mathbb{Q}$, and one must identify the induced ring structure on $H^*(X;pi_{-*}(E)otimes mathbb{Q})$.