// generally useful functions
function type(x) { // does not work in general, but works on JSONable objects we care about... modify as you see fit
    // e.g.  type(/asdf/g) --> "[object RegExp]"
    return Object.prototype.toString.call(x);
}
function zip(arrays) {
    // e.g. zip([[1,2,3],[4,5,6]]) --> [[1,4],[2,5],[3,6]]
    return arrays[0].map(function(_,i){
        return arrays.map(function(array){return array[i]})
    });
}


// helper functions
function allCompareEqual(array) {
    // e.g.  allCompareEqual([2,2,2,2]) --> true
    // does not work with nested arrays or objects
    return array.every(function(x){return x==array[0]});
}

function isArray(x){ return type(x)==type([]) }
function getLength(x){ return x.length }
function allTrue(array){ return array.reduce(function(a,b){return a&&b},true) }
    // e.g. allTrue([true,true,true,true]) --> true
    // or just array.every(function(x){return x});


function allDeepEqual(things) {
    // works with nested arrays
    if( things.every(isArray) )
        return allCompareEqual(things.map(getLength))     // all arrays of same length
               && allTrue(zip(things).map(allDeepEqual)); // elements recursively equal

    //else if( this.every(isObject) )
    //  return {all have exactly same keys, and for 
    //          each key k, allDeepEqual([o1[k],o2[k],...])}
    //  e.g. ... && allTrue(objectZip(objects).map(allDeepEqual)) 

    //else if( ... )
    //  extend some more

    else
        return allCompareEqual(things);
}

// Demo:

allDeepEqual([ [], [], [] ])
true
allDeepEqual([ [1], [1], [1] ])
true
allDeepEqual([ [1,2], [1,2] ])
true
allDeepEqual([ [[1,2],[3]], [[1,2],[3]] ])
true

allDeepEqual([ [1,2,3], [1,2,3,4] ])
false
allDeepEqual([ [[1,2],[3]], [[1,2],[],3] ])
false
allDeepEqual([ [[1,2],[3]], [[1],[2,3]] ])
false
allDeepEqual([ [[1,2],3], [1,[2,3]] ])
false

<!--

More "proper" option, which you can override to deal with special cases (like regular objects and null/undefined and custom objects, if you so desire):

To use this like a regular function, do:

    function allDeepEqual2() {
        return allDeepEqual([].slice.call(arguments));
    }

Demo:

    allDeepEqual2([[1,2],3], [[1,2],3])
    true
    
  -->

function arraysEqual(a,b) {
    /*
        Array-aware equality checker:
        Returns whether arguments a and b are == to each other;
        however if they are equal-lengthed arrays, returns whether their 
        elements are pairwise == to each other recursively under this
        definition.
    */
    if (a instanceof Array && b instanceof Array) {
        if (a.length!=b.length)  // assert same length
            return false;
        for(var i=0; i<a.length; i++)  // assert each element equal
            if (!arraysEqual(a[i],b[i]))
                return false;
        return true;
    } else {
        return a==b;  // if not both arrays, should be the same
    }
}

//Examples:

arraysEqual([[1,2],3], [[1,2],3])
true
arraysEqual([1,2,3], [1,2,3,4])
false
arraysEqual([[1,2],[3]], [[1,2],[],3])
false
arraysEqual([[1,2],[3]], [[1],[2,3]])
false
arraysEqual([[1,2],3], undefined)
false
arraysEqual(undefined, undefined)
true
arraysEqual(1, 2)
false
arraysEqual(null, null)
true
arraysEqual(1, 1)
true
arraysEqual([], 1)
false
arraysEqual([], undefined)
false
arraysEqual([], [])
true

/*
If you wanted to apply this to JSON-like data structures with js Objects, you could do so. Fortunately we're guaranteed that all objects keys are unique, so iterate over the objects OwnProperties and sort them by key, then assert that both the sorted key-array is equal and the value-array are equal, and just recurse. We CANNOT extend the sort-then-compare method with Maps as well; even though Map keys are unique, there is no total ordering in ecmascript, so you can't sort them... but you CAN query them individually (see the next section Option 4). (Also if we extend this to Sets, we run into the tree isomorphism problem http://logic.pdmi.ras.ru/~smal/files/smal_jass08_slides.pdf - fortunately it's not as hard as general graph isomorphism; there is in fact an O(#vertices) algorithm to solve it, but it can get very complicated to do it efficiently. The pathological case is if you have a set made up of lots of seemingly-indistinguishable objects, but upon further inspection some of those objects may differ as you delve deeper into them. You can also work around this by using hashing to reject almost all cases.)
*/

<!--
**edit**: It's 2016 and my previous overcomplicated answer was bugging me. This recursive, imperative "recursive programming 101" implementation keeps the code really simple, and furthermore fails at the earliest possible point (giving us efficiency). It also doesn't generate superfluous ephemeral datastructures (not that there's anything wrong with functional programming in general, but just keeping it clean here).

If we wanted to apply this to a non-empty arrays of arrays, we could do seriesOfArrays.reduce(arraysEqual).

This is its own function, as opposed to using Object.defineProperties to attach to Array.prototype, since that would fail with a key error if we passed in an undefined value (that is however a fine design decision if you want to do so).

This only answers OPs original question.
-->