Yes it's very harmful to inhale the bleach fumes because if inhaled too much the bleach fumes can cause damage to your lungs and organs.
Bleach fumes and bleach can also cause eye and skin damage so it's very important to wear gloves when using and handling bleach ad ventilate the area if possible before using bleach to clean.
Also very important never mix ammonia with bleach as it creates a deadly gas that can be fatal within minutes of inhaling it.