@samalkhaiat is right. Spinors are built on complex numbers.
Speaking of geometrical intuitions:
@OP, what you know about vectors is false. The object you think of as a vector is in fact a spinor. Spinors are quite simple, it's the vectors that are bizzare.
Spinor is an object that carries the following information:
1. Direction.
2. Magnitude.
3. Information regarding number of times it has been rotated (odd/even).
I imagine it as "an arrow attached to a wall". Think of an arrow (say a pencil) attached to a wall using a ribbon. One edge of the ribbon is connected to the pencil, the opposite one is connected to the wall. The geometrical object that can be used to describe this system is a spinor. You have the direction the pencil is pointing, the length of the pencil and the number of wraps of the ribbon (only the parity).
You might have thought that this system (save for the ribbon) is in fact a vector. The concept of a vector in school is taught as something that has direction and magnitude. This is not true. A vector can describe much more. Something with direction and magnitude (plus parity number of wraps) is a spinor. A vector is a much more complex object.
The "parity of wraps" feature of a spinor is a reflection of the fact that certain systems pointing in some directions may be realized in more than one way. Imagine you want to build a robotic arm that can point (say a gun) in any direction. You can build it with solid blocks of any shape and rotating joints. However the joints can rotate only 360 degrees at maximum. They are similar to human joints in this respect.
How many joints do you need to build such an arm? Three. When you have the 3 joints, how many combinations of the joints angles are there to select one direction? There are two. You can encode any direction in two ways. A spinor is a full mathematical description of such a system. You may think of the joints rotations as of the Pauli matrices.
I must say one thing here, that spinors have nothing to do with quantum mechanics. There are purely classical objects. In fact, they need to be quantized to admit properties needed in QM. So we should talk about two kinds of objects: classical spinors and quantum spinors. Unfortunately, spinors are usually introduced in the quantum flavor, without the more simple classical case.
Let's drag the robotic arm analogy a bit further.
The arm is pointing in some direction. Its joints are set to some angle to encode that direction. Now:
1. If the joints have restricted angle range (say 0-360 deg) their rotations can be thought of as Pauli matrices and the setup is described by a spinor.
2. If the joints are unrestricted (they can rotate as many times as you wish) their rotations can be thought of as normal rotation matrices. You can build more devices using a limited count of such joints than with restricted ones and it reflects the fact that vectors carry more information than spinors.
3. If you have joints that can be set only in a discrete angle increments (i.e. with a ratchet), you get quantized rotations. The system is described by a quantum spin vector and the joints angles correspond to quantum spin numbers in certain direction.
As you may have guessed, an electron would correspond to a combination of 3 restricted joints with ratchets.
The most important thing to change in popularization of quantum mechanics in my opinion is the introduction of classical spinors. We should talk more about them, instead of introducing quantum spinors directly.