Hi @Anonymous,
Does the file with the complex component only contain the one part or does it contain multiple parts that you don't want to import into the new design. If the file only contains the one part than all you have to do is just drag the file from your data panel onto your layout in Fusion 360 (assuming that you have new design open). This will automatically prompt a "Move" dialog box to come up.
If the original has multiple components/bodies and you only want to import one of those, you can just right click the component/body from the browser tree and select "Save As" and repeat the steps from above.
Here is a screencast I made showing how to do all of that:
http://autode.sk/1VQA6uu
Hopefully this answered all of your questions. If so, please feel free to mark this answer as a solution, that way others can benefit from this thread as well. If you have any more questions or would like some clarification, don't hesitate to let me know. I'm more than happy to help!
Thanks,
James Youmatz
Product Insights Specialist for Fusion 360, Simulation, Generative Design