Do I need a ground polygon for analog design?
If you really know what you're doing, and:


you really need to cut down the cost (4 layer PCB is not that much more expensive)
you can afford the extra work needed to route the PCB in two layers
you can afford the risk of a few respins that might be needed to achieve satisfying EMC performance


the ground and power planes are not necessarily needed. I mean, it is possible to make a good design without them, but it is much much more difficult.