How you describe the harmony (chords) at a point such as the beginning of bar 4, depends upon what any chord symbols are going to be used for. There are in fact several ways to notate the passage you're describing. (For this answer I'm assuming that you do indeed want an Am chord sounding with all the bass notes in bars 3-4). Here are some options:
- if you simply want all instrumentalists to know that an Am chord sounds throughout bars 3-4, just give them an Am symbol at the beginning of bar 3. Even though the bass line is changing through these bars, the other instruments will keep playing Am. Even putting a single Am marking in the bass part is a useful thing to do here, as the bass player will be aware that the harmony remains the same, even though their bass line changes notes. This is probably the best way to notate this passage if you feel that the bass line is an independent line, which is not integral to how the harmony functions. (Consider this example: the bass plays an A Minor scale, while the other instruments continue to play Am chords - in this case, it would be a nonsense to describe the chords as Am, Am/B, Am/C, Am/D, Am/E, Am/F, Am/G, even though this would technically describe the relationship between the continuous chord and changing bass line!)
- if, on the other hand, you feel that the notes in the bass line are integral to defining the harmony, and so it would be useful for the other instrumentalists to reflect this this in their own choice of chord voicings/types, then yes, you could give the following chords to everyone: Am, Am/C, Am/E, Am/B. This would also be useful in a score, as it shows the overall harmony, created by all parts together. This could certainly be valid for your example; the bass line doesn't feel like an independent "melodic" line, and it's use of repeated notes and regular changes every two beats (ish) certainly make it feel like it has a harmonic function.
- you could choose some kind of half-way point between these two sets of chord symbols. The first six beats of line two all outline an Am chord, as the bass notes are all chord tones of Am, even though the bass is changing the inversion of these chords through arpeggiation, however, the B under the last chord is not part of an Am chord. So, you could use the following: Am / / / | / / Am/B / | Again, I must stress, this only makes sense if you "hear" the Bs in the bass line here as part of the harmony (i.e. you may want them reflected in the chords played with these bass notes) rather than simply non-chord-tones independent of the accompanying chords.
TL;DR: The short answer to the very specific question about the E under an Am chord is that, yes, this is an Am/E (it is an inversion). But, bass players commonly move between chord-tones when creating a bass line to fit with a particular chord, and use non-chord-tones too. So, you could simply use an Am marking here, and consider the bass line to have a certain amount of independence from the accompanying chord.
Personally, I would favour simply using an Am marking, because I simply hear the bass line as an (reasonably) independent line, which is essentially outlining an Am arpeggio under an Am chord before ending on a non-chord-tone which approaches the root of Em strongly by a fourth. If your bass line more strongly emphasised notes which would not be expected under an Am chord, I think this would make it far more important to use slash chord notation. For instance, if the bass line were primarily using non-chord-tones, say a B throughout bars 3-4, rather than just in the last two beats, this would greatly affect the effect of the harmony, and an Am/B would be a good idea in all parts. Or, for instance, if the whole of the line used an E under the Am chord, this would merit using an Am/E symbol, for the same reason.
Lastly, with a nod to @Matt L, are you sure you want a B under Am harmony at the end of the line? I kind of like the "clash" effect you get here, but other notes might work better. If it is what you want, the distinctive sound of this chord in particular, might justify using the notation in the third bullet-point above.