Why can a pointer variable in c be written as p[i] (i is an integer)?

This is the code

#include <stdio.h>
int main(void){
    int a[3][3], *p, i;
    p = &a[0][0];
    for(i=0; i<9; i++) p[i]=i+1;
    printf("%d\n", a[1][2]);
    return 0;
}

In this example, I know the code will print 6. But since p is a pointer variable, how can it be written as p[i]. I guess that p[i] here is the same as *(p+i) and this will points to the ith element in the 2-dimension array. But my question is why are p[i] and *(p+i) the same if p is only a pointer variable?

PS: I know that if we define an array a[10], a can be seen as a pointer constant.

Edited: I think my confusion lies in why the pointer can be indexed here.

I just wonder why this notation is acceptible in C?