###Etymology

[Wiktionary][1] tell us that :

>(*Nom*) Du latin *passus* (« pas, enjambée, marche » ou « trace de pas »).


>(*Adverbe*) De l’usage en ancien français d’ajouter un substantif signifiant « le moindre » après *ne* :

>>Je ne bois goutte. — Je ne mange mie. — Je ne marche pas.

A *"pas"* is a step, the shortest measure of distance. In old french, to negate, we were using *"ne"* with the shortest measure of what we were doing.

> Je ne bois goutte

→ I do not drink, even a single drop.

> Je ne mange mie.

→ I do not eat, even a little piece of bread (*"mie"* is the *inside of the bread*, but underline the special meaning here)

>Je ne marche pas.

→ I do not walk, even a single step.

> L’usage de *pas* s’est généralisé par un cycle de Jespersen et la négation *ne* est pour sa part devenue optionnelle dans le langage courant :

>>Je marche pas dans cette combine. — J’vois pas où tu veux en venir.

<sub>I do not walk in this ploy (I don't accept). — I do not see where you want to go (what you mean).</sub>

So, **in conclusion**, we can use *"pas"* for nearly every negation and ***"ne"* became optional progressively**.


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###General case
To come back to what you are asking :

> However, I wonder if *"ne"* can also be omitted in other structures, notably with *"personne"*.

Yes, you can in all cases when you are **talking**. When you are writing or in more formal context, it is necessary to write/say the full negation.

As @Chop says :

> When spoken, it is also usual to shorten it, just pronouncing the **N** (a sort of a contraction). When speaking formally, *"Je n'suis pas d'accord."* will be better than *"Je suis pas d'accord."*


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###You example :

>For example, *"Personne ne me connaît ici"* means that *"No one knows me here"*, but can I speak *"Personne me connaît ici"* to express the same thing?

*"Personne me connait ici"* will be understood as *"Nobody knows me here"*

  [1]: http://fr.wiktionary.org/wiki/pas