[1] when i say "we" i mean you Americans, i'm in the UK and have the NHS
My wife did this with
* Her cardiologist. He did a _great_ job of heart surgery after her heart attack, but she wasn't thrilled about the way he ran his clinic on her follow-up visits. So she got another guy. That this guy was closer to the house is an unexpected bonus.
* Her orthopedic surgeon. Visited several guys when it came time to replace her knees. Picked one she liked who is also uber-competent.
Digression: Ten years ago when this guy was a middle-aged doctor knee replacements were a hand-crafted affair. Now the doctor runs the operation but a kind of CNC machine does the actual cutting, uses lasers to measure things and customize the new knees.
CNC machines cutting new knees: it's an awesome century so far.
http://www.chooseandbook.nhs.uk/patients/faqs/index_html#4
Also, any time I contact our local GP surgery they always ask if I want to see my usual GP or someone else.
I don't buy that we can't handle the truth and need to be shielded from the pricing and costs structure of medical procedures. Especially because most medical procedures are purely routine matters and the hospital can just publish the average cost of those procedures. You'll then at least get a rough impression of what the price is going to be and you'll be able to compare hospitals based on that.