Under eye filling to address the undereye hollows is considered an advanced technique. This means that the nurse down the street is probably not a good choice for having this procedure. Similarly if your dermatologist tells you that they are not comfortable injecting this area, you need to respect this. You might look for an oculoplastic surgeon in your area with a reputation for doing this treatment.
Regarding complications, the biggest complication is the risk of bruising. Approximately 30% of individuals will have some degree of bruising. Two out of one hundred will have a big black eye that can last up to 3 weeks. Naturally everyone will think your significant other slugged you. Having a good story is helpful. Bar fight usually works for guys. HIding these with makeup is not so significant. Avoiding things that can thin the blood is very helpful to reduce the risk of these. It is important to remember that you are basically getting surgery, it just happens to be delivered by a syringe.
The next most common complication is not getting enough volume. This is often the fault of the injector worried about your pocketbook. Hey, this is expensive service, it is much better to spend a little more and get the volume you actually need. This avoids the disappointment that will be caused if you are still hollow after treatment.
Lumps, bumps, and swelling are also to be expected after treatment. Generally, I like to see my patients back within a week to seen if anything needs adjusting. The genius of this treatment is that it is straight forward adjusting this material manually or with enzyme when needed.
Regarding the risk of some horrible event like blindness or being poked in the eye with a needle, it is critical that you seek treatment from injectors who are comfortable treating around the eye and know their anatomy forward and back. I do midface surgery and know the surgical anatomy of the midface and lower eyelid like the back of my hand. This is very important. There are large blood vessel that must be avoided so that filler is not injected into a vein which might affect the circulation. The choice of injector makes all the difference.