Are there any serious potential complications of urticaria and angioedema?

Depends on cause. The risk of severe laryngeal swelling or throat swelling depends on the cause of your hives and swelling. This risk can be high in food/ drug allergy but may be low some chronic hive disorder. An accurate diagnosis can help stratify risk.
Usually not. Urticaria or hives can be very distressing and itchy, but not associated with complications unless there is vasculitis called urticarial vasculitis. Angioedema associated with hives is also uncomfortable but not necessarily dangerous unless it affects the tongue or throat. Hereditary angioedema where there is no hives can be life threatening due to throat swelling and asphyxiation. See allergist.
Yes. Angioedema (ae) of the tongue may compromise breathing. Ae in the absence of hives may be a symptom of c1 esterase deficiency. In this disease ae of the windpipe can lead to suffocation while ae of the gut can lead to severe pain & unnecessary surgery. Hives & ae may be symptoms of anaphylaxis a life-threatening allergic problem involving breathing, heart and digestive systems.

Related Questions

How do doctors treat urticaria and angioedema?

Antihistamines. Most physicians will treat urticaria with antihistamines. Often this requires higher than standard doses. In severe cases of angioedema, physicians often prescribe injectable epinephrine. Sometimes steroids are used to get the hives under control but often switched as soon as possible. It is helpful to see an allergist to try and identify any allergic triggers.