Unless you need someone else to hear your bass part at the same time as you're laying it down, or your home recording rig doesn't include some kind of monitor system, even if just headphones, then no, a bass amp isn't essential.
In fact, IMHO, it's easier to record bass direct with a good, dry signal, then touch up eq, add a degree of compression and throw some reverb around it to give it depth.
^^
AXES: Fender '81 The STRAT, '12 Standard Tele, '78 Musicmaster Bass, '13 CN-240SCE Thinline; Rickenbacker '82 360-12BWB; Epiphone '05 Casino, '08 John Lennon EJ-160E; Guild '70 D-40NT; Ovation '99 Celebrity CS-257; Yamaha '96 FG411CE-12; Washburn '05 M6SW Mando, '08 Oscar Schmidt OU250Bell Uke; Johnson '96 JR-200-SB Squareneck Reso; Hofner '07 Icon B-Bass; Ibanez '12 AR-325. AMPS: Tech 21 Trademark 10; Peavey ValveKing Royal 8; Fender Acoustonic 90, Passport Mini, Mini Tonemaster; Marshall MS-2 Micro Stack; Behringer BX-108 Thunderbird; Tom Scholz Rockman. PEDALS/FX: Boss ME-50; Yamaha EMP100; Stage DE-1; Samson C-Com 16 L.R. Baggs ParaAcoustic D.I; MXR EQ-10.