By algorithmic complexity, are you referring to the cellular machinery, or something else?
No. It seems cellular machinery would be in the province of genomic complexity — which in turn is governed by algorithmic complexity as the "instruction set" which tells the genomic instructions what to do. In short, the relation of algorithmic to genomic complexity is as "instructor to the instruction set." So to speak.
BTW, while I'm still thinking about it, I found it rather funny that the Szostak YouTube piece used Beethoven's "Ode to Joy" from the 9th Symphony as background music. For there is absolutely nothing in Szostak that could explain "joy," or "ode," or even "Beethoven."