What's the proper way to daven? Eyes open, closed, or looking in a siddur?
The Gemara Yevamos (105b) teaches that one should face his eyes downward while davening, based on a verse in Melachim (1- 9:3).
The Tur and Shulchan Aruch (95:2) also bring down this halacha: one must look down during Shemoneh Esrei.
Sha'arei Teshuvah (95:1) brings the Zohar (Vaeschanan 10), which says, "One must close his eyes during prayer."
Biur Halacha (95:2) cites the Maamar Mordechai (95:2) that states that the Zohar is arguing with our Gemarah; the Zohar said to close them, and the Gemarah said to look down.
Biur Halacha then brings the Chayei Adam (22:11), who interprets that the Zohar follows our Gemara: the only problem is when one looks at other things in the room during the Amidah, but if one looks down, even according to the Zohar, that's acceptable.
He then rules it's best to be stringent like the Maamar Mordechai and keep your eyes closed.
Mishnah Berura (95:5) says if one wants to look in a siddur, that's also acceptable.
However, Sha'arei Teshuvah (96:2) says everyone should determine whether they can concentrate better by heart or with a siddur, and they should do whatever is better for their concentration.
The Magen Avraham (93:1) also rules that one should do whatever is best for concentration.