I asked this on mse, but I didn't get any answers even after a bounty.
I started studying Daniel Huybrechts' book, Complex Geometry and Introduction. I tried to learn as much backwards as possible, but I limited myself to the concepts of almost complex structures and complexizations. I have studied several books and articles on the subject, including those by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier
I have several questions about the concepts of almost complex structures and complexizations. Here are some:
Assumptions and notations: To let $ V $ be a $ mathbb C $Vector space. To let $ V _ { mathbb R} $ be the realization of $ V $, For almost any complex structure $ I $ on $ V _ { mathbb R} $denote by $ (V _ { mathbb R}, I) $ than the unique $ mathbb C $Vector space, the complex structure of which is given $ (a + bi) cdot v: = av + bI (v) $, To let $ i ^ { sharp}: V _ { mathbb R} to V _ { mathbb R} $ indicate the unique almost complex structure $ V _ { mathbb R} $ so that $ (V _ { mathbb R}, i ^ { sharp}) = V $, $ i ^ { sharp} $ is given by $ i ^ { sharp} (v): = iv $, To let $ has i: V _ { mathbb R} ^ 2 to V _ { mathbb R} ^ 2 $. $ has i (v, w): = i ^ { sharp} oplus i ^ { sharp} $

To let $ W $ Bean $ mathbb R $Vector space. To let $ W ^ { mathbb C} $ denote the complexification of $ W $ given by $ W ^ { mathbb C}: = (W ^ 2, J) $, Where $ J $ is based on the canonical almost complex structure $ W ^ 2 $ given by $ J (v, w): = ( w, v) $, To let $ chi: W ^ 2 to W ^ 2 $. $ chi (v, w): = (v, w) $

For every card $ f: V _ { mathbb R} to V _ { mathbb R} $ and for every almost complex structure $ I $ on $ V _ { mathbb R} $denote by $ f ^ I $ as the unique card $ f ^ I: (V _ { mathbb R}, I) to (V _ { mathbb R}, I) $ so that $ (f ^ I) _ { mathbb R} = f $, With this notation the conditions & # 39;$ f $ is $ mathbb C $linear in terms of $ I $& # 39; and & # 39;$ f $ is $ mathbb C $antilinear with respect to $ I $"are reduced to"$ f ^ I $ is $ mathbb C $linear & # 39; and & # 39;$ f ^ I $ is $ mathbb C $antilinear & # 39 ;.

The complexification under $ J $of each $ g in End _ { mathbb R} W $ is $ g ^ { mathbb C}: = (g oplus g) ^ J $i.e. the unique $ mathbb C $– linear map on $ W ^ { mathbb C} $ so that $ (g ^ { mathbb C}) _ { mathbb R} = g oplus g $

To let $ sigma: V _ { mathbb R} ^ 2 to V _ { mathbb R} ^ 2 $. $ gamma: W ^ 2 to W ^ 2 $ and $ eta: V _ { mathbb R} to V _ { mathbb R} $ Cards be such that $ sigma ^ J $. $ gamma ^ J $ and $ eta ^ {i ^ { sharp}} $ are conjugations. (The $ J $& # 39; s are of course different, but they have the same formula.)
ask::

To the $ sigma $Is there an almost complex structure? $ I $ on $ V _ { mathbb R} ^ 2 $ so that $ sigma ^ I $ is $ mathbb C $linear and why / why not?

If yes to question 1 then $ I $ Necessary $ I = k oplus h $ for some almost complex structures $ k $ and $ h $?

To the $ gamma $Is there an almost complex structure? $ K $ on $ W ^ 2 $ so that $ gamma ^ K $ is $ mathbb C $linear and why / why not?
 Note: I think the answer to question 3 is no if the answer to question 1 is no. However, I think that question 3 will be answered positively and with an explanation if the answer to question 1 is yes and the answer to question 2 is no.

To the $ eta $Is there an almost complex structure? $ H $ on $ V _ { mathbb R} $ so that $ gamma ^ K $ is $ mathbb C $linear and why / why not?
 Note: I think the answer to question 4 is no if the answer to question 3 is no.
Observations that led to the above questions::

$ chi ^ J $ is a conjugation, on $ (V _ { mathbb R}) ^ { mathbb C} $, called the standard conjugation $ (V _ { mathbb R}) ^ { mathbb C} $,

$ has $ is an almost complex structure $ V _ { mathbb R} ^ 2 $,

While $ chi ^ J $ and $ chi ^ { J} $ are $ mathbb C $antilinear, we have $ chi ^ { has i} $ is $ mathbb C $linear.

$ k $ and $ h $ are based on almost complex structures $ V _ { mathbb R} $ then and only if $ k oplus h $ is an almost complex structure $ V _ { mathbb R} ^ 2 $

I actually think $ chi ^ {k oplus h} $ is $ mathbb C $linear for almost complex structures $ k $ and $ h $ on $ V _ { mathbb R} $, not only $ k = h = i ^ { sharp} $,