What are the extra courses required to get a core job in ECE?

depends upon your target - If company is using robotics and matlab then they look for skills in matlab/robotics.

I can tell you about embedded as I work in a company which focus on embedded designs. We never hire students with extra embedded course as over time we have realised that all embedded courses are just useless. They teach such an old technology that it takes us time to force freshers to unlearn it. 

We prefer freshers doing embedded during their four years of engineering on their own. It is practice and passion and not the "knowledge" acquired from the training institutes.

(I am representing startups working in embedded. This may be useful if you are looking for job with startups in embedded field)