Let me supplement Alice's nice answer with a few more points. Regarding
there are antidepressants that do not enhance serotonin but suppress it.
That seems to be tianeptine; it is often brought up such discussions as the counterexample to SSRIs being "real" anti-depressants. More recent research on tianeptine suggests it acts on the opioid receptors... and that's not a new line of antidepressant treatment, but rather an old one, potentially sharing the same addiction risks as other opioid-acting medications (e.g. painkillers).
Which brings us to the next point
There are some that do nothing to serotonin at all, but their therapeutic effects are all the same, according to Hirsch.
Surely there are; bupropion is a more widely used (at the least in the US) anti-depressant that is not an obvious SSRI.
So the effect is independent of the chemical composition. That means the chemical imbalance theory is wrong.
That's basically jumping to conclusions (and more than once). The effect is not totally independent of the chemical composition of the medication. An alternative conclusion is that depression, which is diagnosed based on a checklist of symptoms, is a heterogeneous disease with multiple causes producing somewhat similar symptoms (which, by the way, don't all have to be present to get a diagnostic).
Actually almost every individual neurotransmitter that is suspect to be involved in depression, including serotonin, but also catecholamines (norepinephrine and dopamine) have been subjected to experimental manipulation typically via dietary changes (in live humans). You can read more details in my answer on skeptics, but the main pattern seems to be that while we can't cause healthy individuals to get (majorly) depressed by lowering the aforementioned neurotransmitters, in previously depressed individuals (and to a lesser extent in their relatives) such interventions that lower these neurotransmitters do produce more significant mood changes; the most significant changes are seen in patients who are taking exactly the medication that the diet intervention counteracts. (see e.g. Kumar et al.).
Also current anti-depressants are only 30% to 60% effective (depending who reviews the evidence and more importantly whether substituting medications [several times] counts etc.)
So what to conclude from this? The most sensible conclusion to me is that not everything is understood about the neurochemistry of depression, but surely concluding that neurotransmitters are not involved is rubbish.
Almost every researcher who works in this field has some favorite hypothesis that explains better the current non-quite-satisfactory simple explanations based on one neutrotrasmitter or antoher, e.g. a 2008 review in Nature by Krishnan and Nestler said:
It is now thought that acute increases in the amount of synaptic monoamines induced by antidepressants produce secondary neuroplastic changes that are on a longer timescale and involve transcriptional and translational changes that mediate molecular and cellular plasticity.
I doubt that's the ultimate word on the topic (all such narrative reviews have their biases when it comes to the less investigated aspects, no matter how prestigious the venue or the author), but it's clear that the "brighter bulbs" keep looking for more satisfying answers than the monoamine hypothesis (this what's called in the mass media "chemical imbalance"). I think any ultimately useful answer has to account for how the current antidepressants work... which is by acting on neurotransmitters, which in turn does alleviate depression (better than placebo) in some people but alas does not in quite a few others.
What Krishnan and Nestler favor is another fairly circulated notion (but not so much in the mass media!) that a final common pathway for depression exists. But alas, the case is far from being closed on that. Here are a few textbook paragraphs that contrast diabetes (another heterogeneous disease) with depression, which also gets to this final-common-pathway issue, as well as the difficulty in distinguishing cause from effect with individual neurotransmitter systems:
Researchers have compared MDD [major depressive disorder] to other
chronic, heterogeneous diseases such as diabetes
mellitus (Krishnan & Nestler, 2008), which, like
MDD, is projected to increase in prevalence in the
coming decades. [...] Like depression, diabetes can present
with a variety of symptoms, such as polyuria, polydipsia,
polyphagia, as well as with cardiovascular,
digestive, and nervous system dysfunction. Unlike
MDD, however, these symptoms are a product of
a known molecular basis of the disease: a deficit
in the production or utilization of insulin. The
involvement of insulin—due to the lack of the hormone
or to failure of its intended signaling—represents
a final common molecular pathway (FCMP)
in diabetes that unites the involvement of disparate
physiological systems. This FCMP has allowed
researchers to begin to use powerful animal models
to understand the symptoms, risk factors, and
complications of this disorder. In addition, the
identification of insulin and the pancreatic ß-cells
involved in insulin production have led to dramatic
improvements in the prevention and treatment
of diabetes. At this point, however, an FCMP is
absent in the study of MDD; presently, there is no
neurobiological equivalent in depression of insulin
from ß-cells.
Over the past few decades, many molecular
systems have been implicated in depression. [...] Because of the wide array of molecular
systems that are disrupted in MDD, or in which
disruption leads to depression, it has been difficult
for researchers and clinicians to cleanly link molecular
and behavioral systems. Investigators have
noted limitations with all these single-system theories
of MDD; indeed, to date, there is no known
single molecule that is both necessary and sufficient
to explain all the varied combinations of symptoms
that can be present in individuals diagnosed with
MDD. Genetic studies of MDD, too, have not
been definitive in determining consistent risk loci,
due to heterogeneity, environmental factors, and
nosological limitations (Shyn & Hamilton, 2010). Even evidence of successful antidepressant action
is not sufficient to demonstrate the primacy of the
targetted molecular system in the pathophysiology
of MDD. In fact, some researchers have argued
that anomalies in traditional molecules implicated
in the pathophysiology of MDD, like monoamines
or glutamate, are unlikely to be at the etiological
core of depression (Krishnan & Nestler, 2010). This
difficulty disentangling the causal links between
symptoms profiles and molecular systems has historically
led to simplifications and misconceptions
of the neurobiology of depression (Lacasse & Leo,
2005).
A primary challenge for investigators is differentiating
between molecules that are disrupted
as a consequence of depression and molecules
that, when disrupted, are involved in the etiology
of MDD. Demonstrating that a particular
molecule—for example, a metabolite of a neurotransmitter—
is up- or down-regulated in MDD
is not sufficient to claim that disruption in this
molecule causes MDD. Even when molecules
can be causally linked to depression, the heterogeneity
of this disorder prevents researchers from
linking that molecule to all forms of MDD. For
example, showing that interferon alpha (IFN-)
can cause MDD is not equivalent to proving that
IFN- underlies all cases of the disorder. Indeed,
whereas some molecular systems may be strongly
associated with one particular domain of impairment
(e.g., dopamine and deficits in motivation
and behavior), the effects of other molecular systems,
like norepinephrine, may be less specific.
Adding to this complexity is the fact that MDD is
a moderately heritable disorder (Levinson, 2009)
in which preexisting vulnerabilities interact with
stressful life events to increase risk for initial and
recurrent depressive episodes in a manner similar
to the kindling model of epileptogenesis (Kendler,
Thornton, & Gardner, 2000). Moreover, different
symptoms and symptom clusters in MDD are
inherited at different rates. For example, sleep and
appetite disruption have higher heritability estimates
than does psychomotor retardation (Jang,
Livesley, Taylor, Stein, & Moon, 2004).
Reproduced from Chen and Gotlib "Molecular Foundations of the Symptoms of Major Depressive Disorder" in The Oxford Handbook of Molecular Psychology, OUP 2015