That does not resolve your problem, it only shuffles it around.

You are defending the case that personhood begins at conception, and your argument supporting this case is "It's a unique living organism. A life. Different from a skin cell which isn't a unique organism."

If I get a kidney from someone else, and that person dies, according to your own argument, that kidney is a person. These cells are metabolizing and dividing, therefore alive, and they are unique with respect to the rest of my body.

Perhaps you'll argue that the kidney isn't an organism because it has no means of reproducing itself. To that I have two counterarguments:

- The argument implies that infertile humans are not persons.

- Nature contains every horror imaginable, including clonally transmissible cancers [0]. If my body develops such a cancer and it jumps to someone else, then I die, according to your definition my cancer that's colonizing someone else's body is a person.

I won't claim to know what defines personhood, but an obvious prerequisite (in the context of human life) is the existence of a centralized nervous system. If I am beheaded, and my headless body is placed on life support while my brain is destroyed, I, as a person, am dead, even if my body is alive. If my brain is surgically removed and placed on life support in a vat, and allowed to interface with the world somehow, then I, as a person, am still alive (whether such a life is worth living is a different question).

Accepting this prerequisite resolves the chimera problem, the kidney problem, the infertility problem and the transmissible cancer, and it results in the conclusion that a zygote is not yet a person, as it does not have a nervous system.

[0] https://en.wikipedia.org/wiki/Devil_facial_tumour_disease