Brief Answer:
Read below.

Detailed Answer:
I read your question carefully and I understand your concern.

For some of the neurotransmitters such as dopamine, norepinephrine (noradrenalin), epinephrine (adrenalin) it is possible to test their levels and those of their metabolites in the urine. These are the catecholamines, neurotransmitters involved in the "fight or flight", the stress response.
However these neurotransmitters are produced not only by nerve endings but also by adrenal gland cells, they are also hormones. So their values are more representative of their production in the adrenal gland not in the nervous system. Their value in daily practice is mostly in evaluation of high blood pressure, are tested to check for a tumor of the adrenal gland.

If you meant testing neurotransmitters which are responsible for the interaction between nerve cells, responsible for mood disorders such as depression, anxiety, psychosis etc, unfortunately the answer is no, there is no such blood or urine tests which can aid in determining their levels, their action is at a local level between nerve cell endings and can't be measured. Furthermore levels would not tell everything because the same neurotransmitter can have different actions depending on the receptors where it acts, different types of receptors lead to different responses.
So there is no useful blood or urine test on that regard.

I remain at your disposal for other questions.

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