I don't think its really possible to entirely elmininate the Marxist-shaped hole in human history and pyschology, let alone make it unborn with just the death of Marx. But it might be possible to delay it. Like other posters have gone over, there was already extensive socialist movements before Marx and many other fellow travellors in anarchism and radical republicanism, but if there is no Marx and Engels type figures to capture the ziegiest of the moment post-48 and scientifically express socialism's analysis of itself and its environment the way Marx did, then there might be decades before a similar work resounds throughout socialist (and non-socialist) thought the way the Manifesto did. And in those decades all the non-Marxist traditions would not have been idle and would have been productively theorizing and analyzing and dunking on each other with each new moment in history. From here *Marxism might settle in as but one smaller component of socialism as other schools entrench themselves with the advantage of being first and being the ones to shelter and nurture many other thinkers that would have OTL gone on to make Marxism a living political movement in much the same way that Paul and Augustine and made Christianity Christianity.
Yet scientific socialism wasn't thought up completely out of the ether by Marx. Despite his constant stabs at them as utter fools he's destroying with facts and logic, Marxism is incredably influenced by Hegelian Dialectics and Ricardian economics and Marx used their formulations as the foundation for him to go on and disprove them (its where Marx caught the barter myth brain-virus and preceded to make budding communists cringe for the next century reading that section of his work). Before Marx it was Ricardian socialists who were the latest hotness in scientific socialist analysis and before them you had everything from Blanc to Babeuf and from Paine to Proudhon.
A lot of the thinkers and activists that were OTL Marxists and are ATL socialists/anarchists of different schools would still introduce a lot the same scientific principles and economic theories in their new homes, and a lot of what made Marxism put forth its claim as the science of socialism would instead come about in these new centers. Part of the reason the Manifesto blew up in such a huge way was how clearly logical and accessible it was and how its language was such a great forum for later thinkers to interact with each other, but in a world without Marx its perfectly possible for another school to achieve this mandate, and so I think whatover tendancy influences and in turn is influenced by the new wave of people like Kautsky would take up much of OTL Communism's role.
All this is a lot to say that I think what you'd have is this minor and slightly obsurce *Marxism that is all up on this primo Hegelian shit and the specific traditions of logic Marx followed, but the funtional equivalent of Marxist Communism's role and much of the same meat of its economic calculations would be *Communism, the expansion and mutation of another school into Marxism's territory.