###Etymologie

Wiktionary 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.

_{I do not walk in this ploy (I don't accept). — I do not see where you want to go (what you mean).}

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

###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."

###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"

###Etymology

Wiktionary 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.

_{I do not walk in this ploy (I don't accept). — I do not see where you want to go (what you mean).}

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

###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."

###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"