A person may daven shemoneh esrei either with closed eyes, or looking into a siddur, whichever results in greater kavana. But it is forbidden to look anywhere else other than in a siddur; it is brought down that one who looks elsewhere will not merit to see the Shechina when his or her soul leaves this world.
It is brought in sifrei kabala, regarding the possuk “HaShem will fight for you, and you shall be silent ” (Sh’mos 14:14) that the possuk (also) refers to HaShem’s doing “battle” for us up in the Heavenly spheres–that when the prosecuting angel points out that Jews talk in their Shuls, and non-Jews are silent in theirs, HaShem, so to speak, has nothing to answer.
HaShem will do battle for you–against the prosecuting angels and the angels of the non-Jewish nations;
but that is conditional upon
and you shall be silent, during your davening.