Is it customary to apply to your Alma Mater?

[was_promised_cake]

I go to a small engineering school that didn't have a PhD program in my major until this semester. I currently work in the department chair's lab and he has been asking me about applying as a PhD student. I told him that it didn't make sense to make any new applications since I have already been accepted at 6 schools, (2 of which top 12 in the US). He seemed visibly shaken, which is odd since he was the prof who told me to apply to so many schools. Is there something I am missing?

[Anxiously Hopeful]

Congratulations on your acceptances  

It is obviously your choice about where you would like to get your higher education and you don't have to feel obliged to apply to your alma mater.  Having degrees from as many different schools as possible is considered great because that shows your potential and adaptability. So go ahead and enroll in one of the top schools. 

[Jenny01]

I have a feeling based off the little bit of information you have shared that they see a lot of potential in you and do not really want to see you go. Ultimately I do think it is in any students best interest to "leave the nest". Aside from what @Anxiously Hopeful has pointed out it is also an opportunity to build your network find new resources and potential projects. Go on and put that potential to work!!

[Adelaide9216]

I've always been told that it's preferable that students do not conduct their doctoral studies in the same university they did their undergraduate/master's degree. Mostly because it's important to see how your subject is addressed elsewhere and expand your network. I also believe that in terms of scholarships and cv, it looks better. However, in multiple cases, I have seen students doing their undergraduate and graduate studies in the same institution, if for instance, there's a special lab or expertise they want to develop or a specific prof they want to work with and that they could not get elsewhere. It's not bad to remain in the same institution, it doesn't mean that you're setting yourself up for failure, but it's often recommended to see what's being done elsewhere than the institution where you've had your previous degree.

[PokePsych]

It happens, but it is generally (indeed) not recommended. One of the current PhDs in my program (also a first year) stayed at his alma mater but was actually advised to go elsewhere if possible (of course not a program that was a less good fit, etc.). 

[PsyDuck90]

Along with what everyone else has stated, it almost sounds like the chair is hoping you'd stay as a means of keeping/recruiting more talent. Based on what you've shared, you're probably a very qualified applicant and since this program is brand new, they probably want some solid people while they grow and prove themselves. I would recommend you do what is best for you though, despite any pressures that may be coming from your current program.