Question: Is it possible to build a computer (Turing complete) using only diode logic without transistors? I know DTL was a thing, but from what I could tell, they used transistors to amplify signals.

Answer: 
I have been working on a diode resistor gate that I call Light Logic and with a single gate I can create all eight basic gates, Buffer, NOT, AND, NAND, OR, NOR, XOR and XNOR. My project is posted on Hackaday as, [SHEDDING A BIT OF LIGHT ON SOME LOGIC](https://hackaday.com/?s=light%20logic). Not fast but it proves that DRL can do it all if folks do not restrict themselves to signal diodes and resistors. Think out of the box. Very basically a Light Logic gate is a LED coupled to a photo resistor/LDR. This combination acts as a switch much as a NPN transistor. Input 1N914 diodes are wired ahead of the LED and power and output are wired to the LDR much as a DTL gate. Granted the LDR does have a pronounced reaction time but this is a new way to create gates and my goal is a 100 percent transistor and relay free processor. Point, keep stray light from being exposed to the LDR.
This is a tough question. I know that "AND" gates can be made from diodes and that single pull-double throw relays can provide the inversion and amplification. So it looks like it is possible (theoretically)! However, it should be noted that diode logic cannot act as a direct replacement for normal transistor logic in most scenarios due the fact that it uses a path to ground instead of the input being pulled high (or low, as with a PNP transistor). Anyways, good luck!