Alex Christensen focused on stabilizing the build system for the achristensen07/WebKit repository, targeting improved cross-platform development and 32-bit compatibility. He addressed platform-specific build failures by making the bmalloc memory allocator optional in CMakeLists.txt, allowing system malloc to be used across different configurations. Additionally, Alex resolved a 32-bit compilation issue in GResources.cpp by including the necessary mutex header, ensuring successful builds on both 32-bit and 64-bit targets. His work, primarily using C++ and CMake, enhanced the reliability and portability of the build process, reflecting a deep understanding of build system configuration and cross-platform development challenges.