"I do not see anything."
Translation:Hiçbir şey görmüyorum.
It's not a double negative. Hiç is not negative on its own, it's open ended. It's just like English words "any", "ever" etc. These words appear mostly in negative sentences (with another word that makes the sentence negative) or in questions.
Bunu hiç gördün mü? = Have you ever seen this?
Hayır, bunu hiç görmedim. = No, I've never seen this. = No, I haven't ever seen this.
When I was teaching English, a lot of students from certain language backgrounds (especially those that have double negatives) thought "any" and "ever" were negative words. One time, a student thought she heard someone shouting from the next room. She went to have a look and I asked her "Who was it?" and she said "Anybody." I was confused. She meant "nobody" or "not anybody".
This has already been answered above.
The difficulties here are around the fact that the Turkish "accusative case" is not simply an accusative case as used in other languages but rather a "definite accusative case". For this suffix to appear, a word must be both a direct object (accusative) plus specific. Hiçbir şey is by definition non-specific.
Bu şeyi gördüm = I saw this thing. Bir şey gördüm = I saw a thing Hiçbir şey görmedim = I didn't see anything / a thing.
It is outdated and incorrect form. It is similar to writing "alot" in English instead of "a lot" :)
(This is only true when it is an adjective modifying something...if you are wanting to use it as a adjective replacing a noun, it is written separately as " hiç bir" but for all intents and purposes, it is probably better to learn it as one word)