The answer:
I don't know of any chemical system that can maintain the complexity necessary for life at the temperatures you cited (~1,600 C).

Alternatives:
However, a variety of scientists have considered what other chemical systems life might use to cope with different temperature regimes. This reference provides a list of those thoughts along with relevant temperature ranges and other notes of interest:

I reproduced a table from that site here: $$\begin{array}{c|c|} \text{Temp Range} & \text{Macromolecule in Solvent} \\ \hline \text{400° C to 500°? C} & \text{Fluorosilicones in Fluorosilicones} \\ \hline \text{113° C to 445° C } & \text{Fluorocarbons in molten Sulfur} \\ \hline \text{0° C to 100° C} & \text{Proteins in Water} \\ \hline \text{-77.7° C to -33.4° C} & \text{Proteins in Liquid Ammonia} \\ \hline \text{-183.6° C to -161.6° C} & \text{Lipids in Liquid Methane} \\ \hline \text{-253° C to -240° C} & \text{Lipids in Liquid Hydrogen} \\ \hline \end{array}$$

A matter of perspective:
To life evolved on a Methane world and using the lipid - liquid methane system, our (Earthly) life would look like a world of "molten rock" (aka water) and we would look like beings made of liquid rock.

To us, beings using the fluorosilicones in fluorosilicone chemical system would look like beings of molten rock too.