Fertilisation seems possible. It is a specific time that 2 gametes which are not persons become a single cell. This single cell has a continuity with the person born. That is, there is a discontinuity at fertilisation and no discontinuities thereafter, and there is little disagreement that the sperm and egg are not persons.
However embryos split into two sometimes and humans can also artificially split them and continue to split them. Some argue that a new soul occurs when an embryo splits into 2 (which is a somewhat reasonable argument). More problematic is the issue of multiple embryos combining to form a mosaic. Does this mosaic baby have 2 souls, or do 2 souls become 1, or does 1 die (probably not as both cell lines continue in various ways)?
Further, the embryo becomes 2 distinct structures: the baby and the placenta. The latter supports the baby through pregnancy but it is not exactly part of the baby. The embryo can develop solely into support organs (placenta) in a molar pregnancy (hydatidiform mole) and there does not appear to be a person at any stage even though fertilisation has taken place.
The problem with womb being the definition of personhood is what about ectopics that survive? Abdominal gestation occurs rarely but they are clearly babies. Ectopic pregnancies are usually tubal, but can occur elsewhere, and abdominal pregnancies may have started out tubal.
Theologically traducianism implies personhood at conception for continuity. There must always be a soul because the soul is inherited. Conversely, if the concept of the new creation of souls (creationism) is correct, this potentially allows for a gap between conception and personhood. Note that these arguments (traducianism and creationism) are a result of the theology and our theology should be scriptural. So what does the Bible argue for?
Birth seems too late. Many scriptures point to personhood starting before birth, though to argue specifically for fertilisation from the Bible is a little harder. Job talks about the night he was conceived (Job 3:3).
Although previously I thought conception equalled personhood, I am now not so certain. In Scripture life is very clearly connected to blood. Combining this with what we know about human physiology, one could argue for circulating blood: a heart and blood cells. Death should be defined by absence of a beating heart (not brain death). If this is the case then perhaps personhood starts when the heart and blood cells are made: about 2–3 weeks post conception. Note that this is 4–5 weeks after usual dating; pregnancy dates are calculated from last menstruation which is (usually) about 2 weeks prior to fertilisation.
There is also an enigmatic verse in Ecclesiastes that says
As you do not know the way the spirit [ruach] comes to the bones in the womb of a woman with child, so you do not know the work of God who makes everything. (11:5)It is unknown whether ruach should be translated "spirit" or "wind" here, but if it is the former, it is at least possible that God sends the spirit into a fetus in a way that we do not understand.