Which unit should I use for m/z?
TL;DR: According to current IUPAC recommendations, $m/z$ is an abbreviation for a dimensionless quantity.
Use of thompson unit is indeed currently discouraged.

There is an overview provided in Definitions of terms relating to mass spectrometry (IUPAC Recommendations 2013) [1, p. 1516] (reference numbers were updated):

Labeling of mass spectra
The labeling of the $x$-axis of a mass spectrum engendered the most discussion during the creation of this document; however, in spite of a general desire for a better way to label the $x$-axis of mass spectra, there was no broad consensus for any of the proposed changes.
Therefore, this document continues the use of the definitions of the Gold Book [2] and the similar definitions in the Orange Book [3].
The Gold Book recommendation is for the use of $m/z$ as an abbreviation for mass-to-charge ratio, a dimensionless quantity obtained by dividing the mass number of an ion by its charge number [2].
The thomson unit, defined as the quotient of mass in units of u and the number of charges $(z),$ was proposed nearly two decades ago [4], but has not been widely adopted and is therefore not recommended.
Labeling the $x$-axis of a mass spectrum with any unit of mass such as dalton (Da), atomic mass unit (amu), or unified atomic mass unit (u) is strongly discouraged due to the confusion that would result when reporting spectra of multiply charged ions.
The quantity plotted on the $x$-axis of a mass spectrum is a function of both the mass and charge of the ion.
Furthermore, the use of amu in place of u is strongly discouraged in all cases; it has been used to denote atomic masses measured relative to the mass of a single atom of $\ce{^{16}O},$ or to the isotope-averaged mass of an oxygen atom, or to the mass of a single atom of $\ce{^{12}C}.$

Gold Book entry:

mass-to-charge ratio, $m/z$
The abbreviation $m/z$ is used to denote the dimensionless quantity formed by dividing the mass number of an ion by its charge number. It has long been called the mass-to-charge ratio although $m$ is not the ionic mass nor is $z$ a multiple or the elementary (electronic) charge, $e$. The abbreviation $m/e$ is, therefore, not recommended. Thus, for example, for the ion $\ce{C7H7^{2+}}$, $m/z$ equals $45.5$.

For historical purposes the aforementioned letter to the editor regarding the thompson unit [4]:

The ‘Thompson’. A Suggested Unit for Mass Spectroscopists
Recently, there has been a rapid increase in experiments in which multiply charged ions are generated in mass spectrometers. […]
This welcome development has permitted the measurement of molecular weights of compounds having masses in the tens and even hundreds of kilodalton.
These advances exacerbate the problems caused by a common imprecision,
namely, the terms mass measurement, mass range, etc are often used imprecisely when mass-to-charge ratio measurement, mass-to-charge range, etc are intended.
The implicit assumption that the charge state of ions is unity is no longer valid when very-high-charge states are so easily produced.
With this letter come two suggestions.
First, that the longer but more exact term be used whenever it is intended.
Second, that a unit of mass-to-charge ratio be adopted.
After all, this is the quantity which all mass spectrometers measure; mass is a derived quantity requiring an independent measurement or knowledge of charge.
Such a unit would be defined as the quotient of mass, in units of u* and the number of charges, $z.$
The number of charges could be positive or negative, depending on the sign of the charge.
The name Thomson suggests itself in view of J. J. Thomson’s contributions to measurement of this quantity and his preeminent role in the evolution of mass
spectrometry.
Using standard rules for abbreviation, we have $\pu{1 Th} ≡ \pu{1 u/\text{ 
atomic charge}}.$
For example, the molecular weight of the peptide myoglobin (isotopic average molecular weight $\pu{16950 u})$ can be deduced from measurement […] of a peak at $\pu{998.0 Th},$ provided it is known that the ion bears $17$ charges.
If this suggestion is accepted, other simplifications ensue.
For example, the benzoate anion, mass $\pu{121 u}$ and charge $-1$ atomic units, is $-121$ Thomson not $m/z~121.$
The latter is actually the mass-to-charge ratio of the corresponding (unstable) cation!

* $\pu{1 u} = m(\ce{^{12}C})/12 = \pu{1.660540E-27 kg}$
The unified atomic mass unit, $\pu{u},$ is also known by the name Dalton and
the symbol $\pu{Da}.$

References

Murray, K. K.; Boyd, R. K.; Eberlin, M. N.; Langley, G. J.; Li, L.; Naito, Y. Definitions of Terms Relating to Mass Spectrometry (IUPAC Recommendations 2013). Pure and Applied Chemistry 2013, 85 (7), 1515–1609. DOI: 10.1351/PAC-REC-06-04-06. (Free Access)
IUPAC. Compendium of Chemical Terminology (the “Gold Book”). Online Version (2019-) Created by S. J. Chalk, 2nd ed.; McNaught, A. D., Wilkinson, A., Eds.; IUPAC Recommendations; Blackwell Scientific Publications: Oxford, 1997.
IUPAC. Compendium of Analytical Nomenclature (the “Orange Book”), 3rd ed.; J. Inczédy, T. Lengyel, A. M. Ure, Eds.; IUPAC Recommendations; Blackwell Science: Oxford, 1998. https://media.iupac.org/publications/analytical_compendium/
Cooks, R. G.; Rockwood, A. L. The Thomson-A Suggested Unit for Mass Spectroscopists. Rapid Communications in Mass Spectrometry 1991, 5 (2), 93–93. DOI: 10.1002/rcm.1290050210.