I am in the process of remodeling my basement bathroom. My original
idea was to have a non-tiled, dyed, concrete shower floor. The issue I
am having is that I can not find any information on a best practice for
doing this or more importantly whether or not this is even a feasible
solution.

Everything I have come by has to do with installing a pre-fab shower
pan (fiberglass or similar) or constructing a mortar shower pan and
tiling over it.

Can anyone tell me if concrete or cement can be used solely for a
shower floor? If so, I assume I would still want to install a CPE
(chlorinated polyethylene) membrane. Then seal around the drain at the
membrane with silicone or polyurethan and finally build a sloped
concrete base inside the closed pan.

If this is a feasible and practical solution, is there any specific
type of cement or concrete I should use. I definately want to avoid
cracks and my brother-in-law had suggested using an acrylic concrete.

Should any specific curing compounds/agents be used?
Sealents?
Issues to be aware of?

ask your inspector what type of insulation is required in your area.