Chemical Engineering gets you into Mech. Eng. stuff, Elec. Eng. stuff, and Bio. Eng. stuff depending on which direction you head in terms of electives and graduate degree.
I got an undergrad in Chem. Eng. and a Masters in the same. My master's took me towards Electrical Engineering (nanotechnology/Carbon nanotubes/High Vacuum Field Emission). Then I got a job which has twisted my education towards law and more electrical engineering (fuel cells/batteries).
The concepts you learn as a Chem. Eng are very applicable to EE. EE has voltage and current while ChE has concentration gradient and mass flow. Same principles.
BTW, pure Chemical Engineering is generally: Chemical Plant Design, Petroleum, Chemical Manufacturing and Materials (Dupont, Gore, Dow).