Many thinkers, politicians and simply people have tried to answer this very old question. Right-wing authors are content to ascribe the label “Marxist” to Bolshevism for they hope that the crimes and absurdities of Bolshevism will discredit the Marxist theory that they hate for other reasons (its insistence on exploitation or its atheism for instance). Left-wing authors are divided: Bolsheviks and their supporters claim Marxism because they think to find in it a sort of legitimacy. Other left wing authors consider this claim as an imposture.
I think that the answer to our question depends entirely on the definition you give to the term “Marxist”. And it is possible to find reasonable definitions of Marxism which confirm either that bolshevism is Marxist or that it is not. But a definition does not make a theory. So, this approach does not put us on the right way to understanding.
But is our basic question not badly formulated? Is the right question not rather: if Marx had lived in the twentieth century, would he had approved or disapproved the Bolshevik system? My answer is in two stages. Firstly, the new formulation brings us in an exercise of speculation, for we don’t find in Marx’s work elements that unquestionably settle the case. But, secondly, as it is unavoidable, let us speculate.
Marx advocated a classless society. Was Stalinist or post Stalinist communism in Russia and in the satellite countries a classless society? I would answer “No”. Was it on the way to become one? I would again answer “No”. The very interesting book of Michael Voslensky “Nomenklatura” (1984) has revealed to the world the hidden ruling class in USSR: this class aimed certainly not at disappearing, as the socialist State in Marx’s theory.
This is the main reason why I agree with the theory of an imposture. To be sure, it is not excluded that Marx would have approved Bolshevik communism. As a simple human being, he could have been inconsistent. But without more precise information, I would prefer the consistency option.
But after all, is all this so important? The fundamental question must be: is Russian communism a model to follow? The opinion of Marx could be interesting, but not more than yours and mine. And here, the answer needs, I think, no speculation.