Place a strainer over a medium bowl and drain the liquid from the artichoke jar into the bowl. Reserve the liquid and set aside the artichokes.
Add 3 tablespoons liquid from the artichoke jar to a large saute pan over medium heat along with a pinch of salt and a few grinds of pepper. Add the baby spinach in handfuls and stir until all the spinach is wilted, then cook for a few minutes, allowing the liquid to evaporate a bit. Meanwhile, chop the artichoke hearts and add them to a large bowl.
Once the spinach is completely wilted down, add it to the chopped artichokes. Stir in the chive cream cheese, sour cream, 8 ounces pepper jack cheese and a pinch of black pepper and mix thoroughly with a rubber spatula or wooden spoon.
Transfer the spinach mixture to an 8-inch square baking dish and smooth off the top. Top the dip with the remaining 4 ounces pepper jack cheese. Place on a sheet tray to catch any drips, and bake until the edges are bubbly and the top is golden, about 25 minutes. (If desired, broil for 2 to 3 minutes to brown the top further.) Serve with the tortilla chips.