I assume they will have a new batch every day. My guess is that they will need a total of about 500 in order to find enough people who have no hardship for a 2-month sequestered trial, are unbiased, and are willing to advise imposition of the death penalty if appropriate.
Once they find the group with no hardships, they will ask questions about bias and the death penalty. An unlimited number of jurors can be stricken by the judge "for cause" (bias, etc.). They will need to have at least 40 jurors left at that point, so IMO, for this case, they'd better have at least 100 when they start that phase.
Because this is a death penalty case, each party will then get 10 "free strikes" (peremptory strikes) of jurors that they can use for whatever reason they want (except for an unlawful reason, like to kick off all the minorities). Normally, in AZ, the parties get a printout of all the remaining jurors on the panel, and then they pass the list back and forth, each party crossing off one juror on each "turn," until 20 are crossed off.
Then there will at least 20 jurors left. The first 12 will be the jury; the next 8will be the alternates. Normally, the jurors are not told which of them are the alternates until the end of the trial--at that point, if there are still more than 12 jurors left in the box, the remaining alternates will be excused. Although sometimes I've seen judges let the alternates stay for deliberations in cases in which the deliberations are expected to take awhile, in case jurors are "lost" during deliberations (e.g., by being seen passing notes to the attorneys, getting ill, or whatever).
ETA: There may be additional peremptory challenges allowed in light of the number of alternate jurors. HHJP will have some discretion in increasing the number of challenges allowed.