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More precisely, from the classification of real Clifford algebras we see that in dimensions 1 and 5 mod 8 they are actually complex matrix algebras. However this natural complex structure can be easily "seen"( or seen) as follows : the chirality(or volume)element commutes with everything and squares to -1, so the algebra must be complex, no need to see the general classification.

My question is if it is possible to easily "see" the quaternion structure in dimensions 2,3,4 mod 8 without looking to the general classification.

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    Since ${\cal C}\ell(2)\cong\Bbb H$ embeds in ${\cal C}\ell(n)$ for $n\ge2$, they all have quaternionic structure, just not necessarily canonical ones. One can see this as a manifestation of $\Bbb H$ embedding in $M_2(\Bbb C)$ and $M_4(\Bbb R)$. One would want a stronger way of describing $\Bbb H$ as an invariant of ${\cal C}\ell(n)$ for $n\equiv 2,3,4$ mod $8$. A couple other things: (i) while I haven't checked, I doubt $\mathbb{H}\cdot I_n$ is inner-automorphism invariant in $M_n(\Bbb H)$, and (ii) there is no $\Bbb H$ action on $M_n(\Bbb H)$ compatible with their actions on $\Bbb H^n$. – 2017-02-12
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    They all surely have, but I wanted to see it canonically as 1 and 5 mod 8 have canonical complex structure. – 2017-02-12

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