The variation proposed is interesting and the one favoured by the computer. I went further and thought when seeing your variation (d5) that it reminded me of certain variations of the Ponziani, with one single (huge) difference: white has not yet played c3. So, taking on d5 to implement Cc3 came to my mind, with the following variation (as a sample and not definitively concluding - and with computer assistance):
1.e4, e5; 2. Nf3, Nc6; 3. Bb5, d5; 4. exd5, Qxd5; 5. Nc3, Qd6 (for instance); 6. Qe2,Nge7; 7. Nxe5, Be6; 8. f4, 0-0-0; 9. Ne4, Qd4; 10. c3, Qb6; 11. Nc4 etc. With a good advantage to white. Of course, the line first proposed by another intervenant is less "messy" and more clear-cut but I just propose another option. It remains to you to make your choice. The variations we both propose are different in nature. The first respondent proposes you a material advantage in exchange for the bishop pair (for what it is worth without convenient development). My variation also proposes a material advantage but doesn't surrender the bishop pair. Furthermore, my variation keeps the center solid (pawns in the center), somehow avoiding counterplay and hindering free piece play on black's part. The idea of d5 is thematic, of course, but premature and clearly inferior but it has one advantage: you are on your own and one single misstep and problems appear. It could be interesting in blitz play but not more, i think. The positions arising after d5 should be examined by you (and possibly others) with computer assistance to refute your missteps in analysis and to get an idea of how to handle various responses.