I had an interesting idea and decided to take 20 minutes of my time to go through what I call 'The Google Alphabet'.  I simply typed in a letter then recorded the first link that popped up.  I think your results are influenced by your search history, so each Google alphabet should be a little different.  Click 'READ MORE' and get my personal Google Alphabet...

A- Amazon

B- Best Buy

C- Craigslist

D- Delta

E- Ebay

F- Facebook

G- Gmail

H- Hotmail

I- iTunes

J- Jets Pizza

K- Kohl's

L- Lowes

M- MapQuest

N- Netflix

O- Otis

P- Pandora

Q- Quotes

R- Rate My Professor

S- Sears

T- Target

U- USPS

V- Verizon

W- Walmart

X- Xbox

Y- Yahoo

Z- Zillow

The majority of the letters will yield the same search results on any computer... But, their will be a few letters that will be different.  Usually Q,Y, and P for some reason.  I tested this on a few computers around the office, and yes you do indeed have a unique alphabet each time.  Try it out for yourself if you have some time to kill, then post your results here!