I have an old Dtps. Newberry Parfait 'Picotee' that has a new leaf, maybe two starting from a point on the bottom outside of the plant, right were a new root or spike would come from. Is this normal? I'm not concerned about it as its alive and well, I just thought it very interesting and though i would mention it. I'm very busy but i will do my best to get a picture of it in here in the next couple of days.
Yes, they can grow directly from the base, sometimes they will have their own roots and sometimes they will be rootless. If you leave it to grow it should produce it's own spikes and can make a great display if it and the main plant flower together.
Sometimes they form when then main crown is damaged, but sometimes they will form for no reason. As Marion said they can make a great display if they flower at the same time as the parent.