I believe "complicated" is just another term for "complex" which is a cyst that contains both solid and liquid matter. They, as well as other types of ovarian cysts, are almost always benign (non-cancerous).
Ultrasound is not exact so some cysts may appear to be complex when they are not. This website is helpful in understanding cysts - http://ovaryresearch.com/ovarian_cysts.htm. It also states that you should almost never have to lose your ovary for a benign ovarian cyst. A surgeon with good cystectomy skills should be able to remove just the cyst and preserve your ovary and its life long endocrine functions of producing hormones to keep every cell in your body healthy. Yes, the ovaries continue to produce hormones our whole lives. They do not shut down at menopause. The removal of one or both has been shown by numerous studies to do more harm than good predisposing us to all kinds of health problems.
I wonder if birth control pills (although not needed for birth control since you had a hysterectomy) may help keep them from forming.