First and foremost it is important to understand that if this goes to court, the court will decide custody based on one thing...one thing only. "What is best for the child". NOT what is best for mom, or dad, or really anyone/thing else. The court is concerned about the child.
Now, in most cases, the court will find its best if BOTH parents are in the child life...in fact the court presume this. But the court will listen to evidence to determine how to divvy up custody. SO if there is evidence that the child is better off with one parent or another, they will consider this.
As you describe, it sounds like mom recently came to Arizona...less than 6 months ago. This is significant. That is because the law that determines what state has custody will look to where the child has lived in the last 6 months. This law is called the The Uniform Child Custody Jurisdiction and Enforcement Act (UCCJEA). The UCCJEA determines where jurisdiction lies if there is a dispute.
If dad files now, in AZ, it will take custody.
That is not the end all the more important issue than what state has jurisdiction is what is best for the child. So if mom can establish that the child is better off with her, the court will grant that...but this comes back to her reason for travel to CA.
If there is a reason for travel other than “trying to get away from dad” that can be grounds to allow mom to keep primary custody, giving dad visitation. So that is possible. This is OK if there is evidence dad is a danger to the child. But if there is no such evidence, then the court could look at this as an attempt to keep the child away from the father...and leaving the state would not look good in this regard. But if she has a reason for the move, for example to seek employment, to seek schooling, or even to be closer to family...all of these are reasonable and would not likely be held against her if this lands in court.
I would recommend hiring a lawyer. And filing now, in CA, for custody. This gives her the initiative. That forces him to hire a lawyer and fighting this in CA as well as AZ. It may be he does not take the time to try and keep this in AZ.
I will say if he does...if he hires an AZ lawyer and fights to keep this in AZ it is possible, under the UCCJEA the court in AZ grants this and takes jurisdiction. It is certainly possible. But if the goal is to keep mom with the child and the case in CA, I would file this in CA and ask for the court to award custody to mom (and visitation to the dad)
Let me know if you have more questions...happy to help if I can