"proper" is something you really (IMHO) shouldn't ask in the context of music. I'd say, whatever works. Of course, works means "it produces the output" (sound) you want, and/or that others expect.

"proper" in the sense of "orthodox" is only using the fingers. Actually, I don't know how you'd use the thumb. I presume you're not "slapping" with it, because that's really not the same as picking with the fingers.

An anecdote: one of the greatest bassist, Jack Bruce (of Cream fame), apparently only uses one finger for picking (is that even the "proper" term? ;-)). Saw it in a concert on TV once. Weirdly, I had learned to use both middle and index finger in an alternating pattern shortly before that. Was pretty (pleasantly) surprised.

Now I can't stop wondering how you actually play bass. Methinks using the thumb would require having your hand in a pretty "unusual" ("improper"/"unorthodox" ;-)) position.

I'm no officially trained reference by any means,
but sometimes I play using my thumb, index and middle fingers: a bit like fingerpicking on a guitar.
The intro (@ 0:49) of this song is played like that.
I see no way how that would be a bad technique to learn. :)

About the proper finger technique: anything goes I'd say:

• try with your thumb fixed on one of the pickups: on the one near the fingerboard gives a fatter tone than on the one near the bridge
• on the beginning of the fingerboard: even more fat
• on the string above the one you're picking
• even without fixing your thumb onto anything

Have fun!

I would suggest that most of the time you want the sound of the bass to be consistent across the strings, which typically means playing them in the same way. In this respect it's a little different from a guitar where you may want the low strings and high strings to sound quite distinct.

Consequently, for learning, I would practice playing regular fingerstyle using your first two fingers if you want that solid sound or a plectrum if you want more of a twang to it. You'll be in fairly technical ground before you actually need to play in the way you describe and then you will probably find another way to work it.

Get solid basics, I would suggest, before going too far into building on them...

I'm only a guitarist, not a bassist, however, I've heard it said that a good bass player never uses his thumb (except to slap of course). But I know what you're saying, in fact whenever I mess around on bass, my tendency is to only use my thumb.