What is the most natural new physics one can expect at the TeV scale: new (supersymmetric)particles or some new (non-commutative) spacetime structure? Up to now, nothing else than one Standard Model (SM) Higgs boson-like resonance has been found at the LHC while many predictions based on effective theories using supersymmetry require several Higgs scalars and needs an entourage of sparticles close in mass to tame its quantum instabilities (I borrow more or less from James D. Wells). 
On the other hand, the spectral and almost-commutative extension of the SM by Chamseddine and Connes, expects only one Higgs boson without other particles in the TeV range. In this noncommutative approach  spacetime appears as the product (in the sense of ﬁbre bundles) of a continuous manifold by a discrete space and it has been proved by Martinetti and Wulkenhaar  that under precise conditions, the metric aspect of ”continuum × discrete” spaces reduces to the simple picture of two copies of the manifold. 
Could it be that this picture of a two-sheets spacetime helps to overcome the technical naturalness issue related to the standard model Higgs (replacing temptatively a low energy supersymmetry by a new geometric framework) and has to be taken seriously in order to progress in the understanding of physics beyond the SM? 
To ask it differently:
Reminding that the Standard model like Higgs boson is a natural consequence of the noncommutative geometric framework, could it be that the discreteness of space-time usually expected at the Planck scale from quantum gravity  already shows up at the electroweak scale through the very existence of the already discovered Higgs boson?
(this formulation could require a new, yet to be defined, heuristic meaning for the term: naturalness)
Last but not least, it is worth noting that to postdict the correct mass of the Higgs boson detected at LHC8, the last version of the spectral model relies on a weak coupling with another scalar that shows up "naturally" in the spectral action just like the Higgs. This "big brother" from the Higgs boson is expected to acquire a vev generating a mass scale above $10^{11}GeV$ for right-handed Majorana neutrinos. It could thus be responsible for a type I see-saw mechanism explaining the neutrino phenomenology beyond the minimal SM. 
May be noncommutative geometry can help to make effective theories more alive and kicking!
In memoriam Ken Wilson
To celebrate the 4th of July "IndependentHiggsday", I wish a happy birthday to the lightest scalar field of the Standard Model and I congratulate experimentalists who work hard to prove physics is alive (and not ordered by theories ;-)!
EDIT : The title of the question has been changed in an attempt to improve clarity (after reading The Higgs: so simple yet so unnatural); the former title was: 

Doubling the number of elementary particles or "doubling space-time"
  to accommodate Higgs boson phenomenology at 8TeV?

 A: The paper you link to contains a model which is simply the Standard Model coupled to a singlet scalar. Its hierarchy problem is just as severe as that of the Standard Model, and as such it is a highly unnatural theory. (To be clear: what I mean by "unnatural" here is that the theory has quadratically divergent corrections to the Higgs mass, and as such the low-energy physics is highly sensitive to unknown ultraviolet parameters. As far as I can determine the Chamseddine/Connes scenario does nothing to tame the ultraviolet problems of quantum field theory.)
A: Notice : This is another tentative answer to address (better than in my former one) the naturalness problem raised by the present stalemate for traditional perturbatively renormalisable Susy-Yang-Mill-Higgs quantum field theories in the LHC phenomenology. It can be formulated briefly as an "educated" guess blending some condensed matter physics intuition and non-commutative vision :
The fine structure of spacetime at the electroweak scale could act as a non-commutative protectorate, ensuring a non-commutative safety mechanism that protects the low mass of the Higgs from diverging quantum corrections up to the Planck scale, just like masses of spin 1/2 fermions are protected by "chirality" and spin 1 bosons are protected by gauge invariance.   
That would explain why SUSY predictions in the conceptual framework of perturbatively renormalisable quantum field theory on the usual four dimensional commutative space-time fails because probing the physics of the Higgs to go beyond the Standard Model probably requires not only to go over much higher energies but also to work with the proper spacetime framework. To make a crude comparison : analysing sound waves with the effective theory of hydrodynamics cannot help the physicist to uncover the atomic structure of matter but listening to the insight gained by the chemist who understood conceptually how to shape matter may help ... 
Moreover progress in the understanding of how non-commutativity could modify the renormalization group flow for the Higgs couplings is already underway.
Then maybe once the proper non-commutative constraint is correctly implemented in physics model-building, one will witness a new increase in "the price of shares of stock in Quantum Field Theory" to quote Weinberg. After all, the Standard Model emerged in the 70s taking seriously non-Abelian gauge groups envisioned in the 50s and thanks to the conceptual understanding of asymptotic freedom in chromodynamics, I remind that this last part is still conjectural today! Then it would be quite natural to go beyond the Standard Model in the 2010s thanks to some new non-commutative geometric ideas developed in the 90s that explain the quantum spontaneous breaking of electroweak symmetry imagined in the 60s!  
Of course this kind of hypothetical heuristics and epistemological retro-analysis is definitely not a technical answer and I could understand that such a speculation is not suited for Physics SE (I am ready to remove it if required). Adding an "epistemology" tag would have helped but there were allready 5 of them.
Remark : I found interesting and inspiring ideas about renormalization in the non-commutative context (but very different from the Higgs) in this article by V. Rivasseau.
A: The  question is hard to answer not because of its colloquial character but because it tries to establish a comparison bewteen two predictions at the TeV scale sustained by two very different theoretical frameworks :


*

*the prediction of supersymmetric particles is made in the context of quantum renormalizable theory with fields interacting in a Minkowski 4D space-time;

*the existence of a fine structure (two-sheets) of spacetime comes with a spectral action principle on an almost-commutative geometric setting.


Despite this fundamental difference, I think it should be possible and interesting to compare them, debate about their mathematical and observational consistency as two effective theories at the TeV scale. I understand effective theories in the modern viewpoint very pedagogically explained by Matthew Schwartz in this lecture note. 
I think it's worth emphasizing that the technical naturalness issue of the Standard Model Higgs boson exists only if one assumes that it is embedded in a larger renormalizable theory that goes along the line of conventional QFT on Minkowski 4D space-time(I would appreciate to be corrected if I am wrong on this statement). 
Insofar as the spectral action principle, applied on a crude almost-commutative geometry and with a Planck-scale cut-off, already proves to be able to deliver the Einstein-Hilbert and the Standard Model Yang-Mill-Higgs terms, it is not unreasonable to expect that noncommutative geometry offers another embedding of the Standard Model to some kind of UV completion with degrees of freedom that would be different from customary QFT in 4D Minkowski space-time. This recent article(2013) for instance goes along such lines proposing a different noncommutative structure which mixes spin and gauge degrees of freedom in a very specific way.
Nevertheless if one want to stick on the naturalness problem of the electroweak scale it could be interesting to reassess in the light of recent developments the former tentative to look at noncommutative geometry as another alternative to compactification(1999). Indeed a almost-commutative toy model on a two-sheeted spacetime with a gravitational and U(1) gauge fields was proved to give a Randall-Sundrum potential for the Higgs field with the correct exponential term to reduce the natural scale of the electroweak symmetry breaking from the Planck scale to the TeV scale without fine tuning.
I quote :

We notice that the non diagonal elements of the matrix algebra, and of
  the Dirac operator, have a twofold interpretation. On one side they
  are the Higgs ﬁeld in the gauge setting, whose natural scale is the
  electroweak scale. On the other side they appear as the discrete
  component of the Levi–Civita connection, with a natural gravitational
  (Planck) scale. It is this dual role which solves the hierarchy
  problem in this setting

It would be interesting to investigate if the exponential term appearing in the 1999 article has some connection with the new singlet scalar field proposed by Chamseddines and Connes in 2012 to postdict the correct mass of the physical Higgs boson observed at LHC8 ...(beyond the mere coïncidence of the sigma labelling for the new degree of freedom in both papers, just a convenient notation for a generic singlet scalar field I guess)!
Comment about the bounty: 
The bounty period beeing over, I thank Matt Reece for proposing an answer but I can't accept it for the very reason given by Peter Shor in his comment. I thank also Mitchell Porter to make constructive remarks. I feel free now to propose my own answer. 
