In this article, we propose a space-time Multi-Index Monte Carlo estimator for a one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. We compare the complexity with the Multilevel Monte Carlo method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method i) has suboptimal complexity of $\varepsilon^{-2}(\log\varepsilon)^2$ for RMSE $\varepsilon$ if the same spatial discretisation as in the MLMC method is used, ii) has the same optimal complexity as MLMC of $\varepsilon^{-2}$ if a carefully adapted discretisation is used, and iii) does not necessarily fit into the standard MIMC analysis framework of Haji-Ali et al. (2015) for non-smooth functionals. Numerical tests confirm these findings empirically.