UPDATE3 In reply to:
"why we can deduce, ... that spinors can exist even in a three (odd-dimensional) space"
Do we ever deduce that? Or do we define it?
Respectfully: I think you're still thinking about these objects as being "in" the space they're associated with.
Spinors live in a spinor space, even when they refer to something in another space.
Try looking at it this way: I would agree that we can describe e.g. vector fields in which there is a 3D vector associated with each point in, for example, physical (3D) space. But the vectors are still elements of a vector space, they are not part of the real space even though we assign their position and coordinates using physical space measures.
However, I see no reason why one couldn't define vectors of any number of dimensions associated with a space of any other number of dimensions, without the coordinates necessarily measuring aspects of the space with which they are associated.
(Forgive me if this isn't maths so much as pedagogy): Imagine, if you will, a fine 2-dimensional lattice (not quite a continuous space) representing the surface of an agricultural zone on planet Earth. At various points we can identify farms (by, say, the lattice-location closest to their farm office) with different combinations of produce, and if there are, say, 57 varieties of produce across all the farms, then we can associate a 57D vector, one per farm, with some points in the 2D lattice...
UPDATE2 (in reply to modified question)
Your identification of the usual complex numbers as being themselves spinors, while not widely recognised as such AFAIK, is consistent with the views of the estimable David Hestenes who has done much to bring Geometric Algebra to the attention of physicists.You might do well to start with http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf in which is demonstrated more clearly the spinorial character of your exp(iθ/2), using a bilinear transformation of vectors.
UPDATE in reply to Comment below from OP (and written prior to OP's revision of original question):
Are you thinking of complex numbers or spinors as being "in" the manifold? it might be better to think of them as simply associated to it, but not "living inside" it. For example, a quaternion (which some recognise as a form of spinor) describing a rotation in R3 is (in my limited understanding) an element of a quaternion-space that happens to be able to describe a particular rotation, but is not itself "in" R3.
There was an earlier comment suggesting that mathematicians had moved on from concern with spinors, and I guess the commenter was pointing in the direction of Clifford Algebras and beyond.
So, if you haven't seen them already, you might want to look at the ever-expanding Wiki pages on Clifford Algebras and the one on Spinors. Though they refer to Reals, it's still not clear to me that spinors can be described without at least implicitly involving complex numbers. For example: the Spinor wiki page says:
"Thus the (real) spinors in three-dimensions are quaternions_"
...yet the quaternion unit vectors i, j, k obey relations like i^2 = -1 (referring now to a pi-rotation).
The Reals, R(1), considered as a primordial Clifford Algebra, sit at the top of a pyramidal table of Clifford Algebras (below which sit the Complex Numbers C(1), together with R(2), and another level down the quaternions, H(1)), so it seems clear that not all Clifford Algebras admit spinors, but I leave it to someone more expert to say just which ones definitely don't admit spinors.
If you find the Wiki pages hard-going, another route in is via the late Pertti Lounesto's wonderful compendium, "Clifford Algebras and Spinors" (London Mathematical Society/Cambridge U. Press), which starts at an elementary level, though it's now quite expensive and may be hard to get.
ORIGINAL REPLY (To initial version of question since edited by OP)
How have you reached the conclusion that "spinor geometry is independent of the existence of complex numbers"?
I am likely at the same level as yourself in pursuing such insights (though trivially elementary to mathematicians - except, famously, to Sir Michael Atiyah).
However, in my limited physicist's understanding, spinors are indissolubly linked to some sense of rotation/angle; and complex numbers are, in geometric terms, about manipulating changes of angle (rotation), so I would be surprised - and also interested to learn - if spinors can be encoded without complex/hypercomplex numbers, either explicitly or implicitly.
