This P-8/737: The Navy was calling it a Multi Mission Maritime Aircraft in earlier press releases. It might a mistake to think of it -- or at least all P-8 variants -- as a narrowly specialized ASW platform.
Also, in addition to its nose radar, I read somewhere that the P-8 -- or some P-8's -- may be getting a ventrally mounted electronically scanned radar antenna looking to the left and right. ( This would be a narrow bulge several meters long behind the weapons bay, which has apparently been moved from behind the wing to in front.)
This radar would give the P-8 a functional capability similar to what the JSTARS aircraft does. This type of radar operation implies higher cruising altitude -- in other words, turbofans instead of turbprops.