Organism is a term that may seem intuitively simple but in reality is rather umbiguous In my humble opinion an organism should simultaneously display a combination of physiological and morphological continuity basically being in one physical piece and having its parts mutually influencing each other but there is a considerable grey area For example males of some fish loose all but sexual organs and literally become part of the body of a female completely fused with the female blood system etc Such a hybrid creature may be considered as a single organism it has one supply of blood etc  In my understanding unseparated Siamese twins have one organism but may have individual personalities It is hard to judge it otherwise because in some cases one of the twins is represented only by an underdeveloped body part Sometimes it is difficult to claim whether a virus a parasite or a symbiont that are present in their hosts are parts of the host organism In fact our genome is stuffed with the remnants of viruses and our every cell has symbionts mitochiondria A zygote could be just a zygote It may or may not be an organism by itself Since we know that the cells within the same organism may have different DNA see some examples above than a zygote of placental mammals may be just a cell albeit a very special cell of the mother organism A freeexisting zygote of fish would be a different story The fact that under some conditions a zygote could develop into a fullsize individual animal is not absolutely unique Sometimes the cells that we call somatic could do it too naturally in some species in artificial conditions  in the others By the way egg cells of some animals could develop into fully grown individual creatures so the question of whether eggs are organisms is just as valid as whether the zygote are