Can I mix anything into cheesecake batter?
There are plenty of recipes that do. You don't want your added ingredients to be too wet, and if they're strongly coloured the colour is likely to bleed into the mix whatever you do (e.g. blueberries). 

The results will be different for baked cheesecakes and chilled cheesecakes, but in both cases it can work. I suggest you keep trying to find a recipe that adds them to the mix, a baked one as that seems to be what you feel like making. Or just try it. Here's a recipe with blueberries to get you started (not tested). 

For chocolate chips specifically, you can just stir in the desired amount after mixing and before pouring it onto the base. They may sink though, depending on your recipe. But this isn't too bad. Citrus zest can also be added to taste (but it needs to be fine). 

If you add a lot of weight or bulk, your cooking time might increase a little.