Many chemicals are cleaners. Just because two have that in common does not mean they are the same. An apple is red. A fire truck is red. They have that in common, but you can’t make a pie out of the latter.
Since bleach and ammonia are, in fact, two very different chemicals, they produce a ‘chemical reaction’ when combined.
Depending on the proportions of each you mix, you can produce different chemical reactions with different results, such as chlorine gas, or an explosion:
2 parts Bleach to 1 part Ammonia:
2(parts) NaOCl + 2NH3 --> 2NaONH3 + Cl2.
That “Cl2” on the right is chlorine gas. It was used as a chemical weapon during World War I and later by Nazi Germany in World War II. If you inhale it, it can be fatal.
Another potential reaction, which occurs when an even greater amount of bleach is added than ammonia, is this:
3NaOCl + NH3 --> 3NaOH + NCl3
That's sodium hydroxide and nitrogen trichloride. Nitrogen trichloride is a very toxic chemical to humans, and even if you did get close enough to ingest it, it would probably explode in your face first, as it is also a very volatile explosive. There is little necessity in explaining why that is bad.
Both results, and a couple other possibilities, are highly unstable and likely to kill you if you don’t know what you are doing. Therefore, it is best not to try to do anything with it. There is a reason those chem labs have fume hoods, etc.