C++ multidimensional array operator

Nope, that is not possible. There are two alternatives, though:

You can have operator[] return an array of a smaller dimension (For a 3D array, it will return a 2D array, for a 2D array it will return a 1D array, and for a 1D array, it will return a single element). Then you can "string them together" with the syntax you want. (arr[x][y][z])

Alternatively, you can overload operator(), because that can take multiple arguments.

Then you can use it like this, to index into a 3D array for example: arr(x,y,z)

But you can't overload [][] or [][][] as a single operator.

Not directly, but you can achieve the same functionality overloading operator[]() and having it return something that supports operator[]() itself.

For example:

class A {
  std::vector<std::vector<int> > vec;
public:
  std::vector<int>& operator[] (int x)
  {
      return vec[x];
  }
};

would allow you to write:

A a;
//...
int y = a[1][2];

because a[1] returns a std::vector<int> to which you can apply operator[](2).

You need to overload operator[] and make it return a new class which only has another operator[].