In a large mixing bowl, beat cream cheese until smooth and creamy.
Add sugar and vanilla, and beat until combined.
Add eggs, one at a time, beating well after each addition.
Beat in rum extract, nutmeg, and eggnog.
Pour slightly more than half the batter over crust - eyeball it! Just slightly more than half.
Carefully place springform pan in the refrigerator, and chill for about 30 minutes.
To the remaining cheesecake batter, beat in pumpkin puree, pumpkin pie spice, cinnamon, and clove.
Carefully pour batter over the eggnog cheesecake layer.
Bake at 325 degrees for 70-80 minutes, until edges are lightly browned, and until the center appears mostly set when jiggled. Take the cheesecake out of the oven when about 1.5-2 inches in the center are still slightly jiggly.
Remove to a wire rack to cool completely.
Once cooled, run a knife around the edges of the cheesecake to loosen, and release the edges of the springform pan.
Transfer cheesecake to a serving plate.