Relativistic CMB

If we are in a hypothetical space ship, at 99.9999 % (or so) of light speed
(as observed from Earth), then Special Relativity tells us that time will
run very slow for us w.r.t. Earth time.

Of course, we don't notice that, but we WILL see that the stars in front of
us seem to come at us at breakneck speed, thus showing a high blue-shift. We
observe that Alpha Centauri is only 10 light minutes away, and coming at us
at close to light speed. Sure enough, ten minutes later it zips by and
disappears in the deep red-shift with the stars behind us.
Very soon we will be out of the Galaxy.

Now we have some time to do some experiments.
Lets try to measure the Cosmic Microwave Background.
What do we see ?

(1) It is still the 2.7K omnidirectional background that we see here on
Earth ?
(2) Or will there be a higher temperature measured in 'front' of us, and
lower 'behind' us ?
(3) Or, since our time runs so much slower than on Earth, will be see a
higher temperature all around us because the Big Bang will seem to have
happened only a short while ago ?

And what are the consequences for our theories of Special Relativity and the
Big Bang and the fabric of space and time if a fast traveler makes one of
these three observations ?

Rob