Is the expense for developing a patent a capital expense? Would it still be a capital expense if I acquired it vs. developed it?

I believe the answer you are searching for is yes.

If you develop something to become a patent, that expense can be capitalized.

If you go out and just buy it, the money you used to buy it can be capitalized.

Yes the expense is of the capital nature.
As we all known to the fact that there are basically two types of expenses which are :
1. capital nature expense

Which helps the business to take benefit of such expense over a long period

2. revenue nature expense

Which are incurred by the business for day to day running of the business. That is for the operating activities of the business.

Thus, the expense for patent is of capital nature. Irrespective of the fact whether itself developed or purchased.