Ask your local contractor.

Depends a lot on local norms-- if the local concrete guys have and routinely use the taller forms, not much beyond the cost of material and excavation. If they don't, then the sky's the limit. You need to talk with a local contractor.

(it makes a huge difference in the usefulness of the space.)

So the basement floor would be lower in the addition? You'd have to step down into it? Or is the first floor level of the house going to be higher in the addition and you'll have steps up into it? If the basement floor is lower, there will be possible problems with footings, drainage and utilities.

Bag o worms. An expensive bag o worms.

I was thinking of lowering part of the basement. We have one part where utilities etc come in and then about 700 sq ft where I was hoping to lower it when I extend that

As rwiegand says, it's doable. In addition to working around existing footings and stepping new footings down, you'll have to work out new footing drains, tie in to existing and install a new sump. Hopefully you don't have a high water table. Try to keep new footings in same coursing as existing block, if it's a block foundation. Your walls may need to be thicker or have more rebar.