Existences of Majorana spinors in $\rm Spin(4)$ and $\rm Spin(1,3)$ $\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$
We know that
$$
\Spin(1,3)=\SL(2,\mathbb C)
$$
and
$$
\Spin(4)=\SU(2) \times \SU(2).
$$
It is also said $\Spin(1,3)$ is the complexification  of $\SU(2)$.

*

*My question is that why there is a Majorana spinor (4 component Real spinor representation?) in $\Spin(1,3)$? But there is NO Majorana spinor (4 component Real spinor representation?) in $\Spin(4)$?

See this fact from the Table https://en.wikipedia.org/wiki/Spinor#Spinors_in_representation_theory

Refs from google search:
http://www-personal.umich.edu/~williams/notes/spinor.pdf
http://scipp.ucsc.edu/~haber/ph251/Spinor_Shijun
https://en.wikipedia.org/wiki/Complexification_(Lie_group)
 A: A Majorana spinor is a spinor for which the Dirac conjugate spinor $\psi^\dagger \gamma^0$ is equal to the Majorana conjugate spinor $\psi^T C$, i.e. $\psi$ satisfies the Majorana condition
$$
(\gamma^0)^T \psi^* = C^T \psi.
$$
Here $T$ stands for transposition, $*$ for (complex) conjugation, $\dagger$ for conjugate transposition, and $C$ is the charge conjugation matrix, which maps gamma matrices to their transposed cousins
$$
C \gamma^\mu C^{-1} = \pm (\gamma^\mu)^T.
$$
From this and the defining relations of the gamma matrices, it is possible to derive details on the form of the possible charge conjugation matrices and the complex conjugation properties of the gamma matrices. With that information we can then check whether the Majorana condition can have solutions. The upshot is that this derivation depends on the signature of the spacetime, and that the Majorana condition turns out to have a solution for signature $(-+++)$, but not for signature $(++++)$.
For the details of this derivation you can look at e.g. the notes in section 3 of https://arxiv.org/pdf/hep-th/9910030.pdf. For an intuitive hint why the existence of Majorana spinors depends on the signature, consider that if we have a set of gamma matrices $\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$ for signature $(-+++)$, we can use $\tilde{\gamma}^0=i \gamma^0$ and $\tilde{\gamma}^j=\gamma^j, j=1,2,3$, as a set of gamma matrices for signature $(++++)$. While things look similar, the appearance of an extra $i$ certainly can (and does) have consequences when looking at an equation like the Majorana condition, involving complex conjugation.
