Reduce oven temperature to 300°F (150°C). In a large bowl, combine cream cheese and sugar. Mix until combined, and then add flour, sour cream, and vanilla. Mix until just combined.
Add eggs, one at a time, and mix after each addition. Reserve a ½ cup of cheesecake batter and pour the remaining batter into the crust.
Divide cheesecake batter into 4 small bowls. Use food colouring to colour each amount of batter a different colour. I chose pink, purple, yellow, and blue.
Pour each colour randomly all over the top of the cheesecake. Use a knife to gently swirl the colours together. Don't overmix.
Cover the outside of the pan with aluminum foil then tie a crockpot liner bag or chicken bag around the springform pan. Make sure the bag goes up the sides of the pan. This will stop water from seeping into the cheesecake.
Place springform pan into a larger pan, and fill the outside pan with water. Fill until it goes half way up the outside of the springform pan. Bake for 1 hour and 15 minutes.
Turn the oven off and leave the cheesecake in the oven with the oven door closed for 30 minutes.
Crack the oven door open and leave the cheesecake in the oven for another 30 minutes. Slowly cooling the cheesecake like this will help keep the cheesecake from cracking.
Take the cheesecake out of the oven and take the springform pan out of the water bath. Cover the top of the springform pan and cool in the refrigerator for at least 6 hours or overnight.