It took me a while to find this: http://www.pnas.org/content/94/4/1054.full
Anyway by Friedlander and Iwaniec (1997). They proved that there are infinitely many primes of the form x2+y4. They mention near the end that they do not have a proof for primes of the form x2+y6 but would like one. So there is a way to go to settle x2+1.
FYI, what I did (not remembering title, authors, anything but the result) was write a program to give the primes x2+y4 and put the first dozen in Sloane's sequence site search feature.
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