const obj1 = {
  a: '1',
  b: 'this is the letter b',
  f: {
    foo: 'what is a foo anyway',
    bar: [1, 5, '3', 4]
  }
};

const obj2 = {
  a: 1,
  b: 'this is the letter b',
  f: {
    foo: 'what is a foo anyway',
    bar: [1, 5, 3, 4]
  }
};

function equal(a, b, mode) {
  const eq = mode === 'strict' ? a === b : a == b;
  if (eq) {
    return true;
  }

  if (a && b &&
    typeof a === 'object' &&
    typeof b === 'object' &&
    Object.keys(a).length === Object.keys(b).length) {

    for (const key of Object.keys(a)) {
      if (!b.hasOwnProperty(key) || !equal(a[key], b[key], mode)) {
        return false;
      }
    }

    return true;
  }

  return false;
};

console.log('objects are equal: ' + equal(obj1, obj2));
console.log('objects are strictly equal: ' + equal(obj1, obj2, 'strict'));