Spectrum is an extremely useful tool for data analysis. Physically, spectral analysis is to change time domain data to the frequency domain. Traditionally, this was achieved through convolutional integral transforms based on additive expansions of a priori determined basis, mostly under linear and stationary assumptions. For nonstationary processes, the best one can do is to use the time-frequency representations, in which the amplitude (or energy density) variation is still represented in terms of time. For nonlinear processes, the data can have both amplitude, intra-mode and inter-mode frequency modulations. Those amplitude variations in a data could come from two different mechanisms: linear additive or nonlinear multiplicative. As all existing spectral analysis methods are based on additive expansions, a priori or adaptive, none of them could represent the multiplicative processes. While the adaptive Hilbert spectral analysis could accommodate the intra-wave nonlinearity, the inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling, phase lock modulations, is left untreated. To resolve the multiplicative processes, we have to resort to additional dimensions in the spectrum to account for both the FM and AM variations simultaneously. The true spectrum should be a full information spectral representation designated as the Holo-Hilbert Spectral Analysis (HHAS), which is one possible such high dimensional spectral representations that would include all the processes, additive and multiplicative, intra-mode and inter-mode, stationary and nonstationary, linear and nonlinear interactions. Indices are also introduced to measure the inter-mode nonlinearity. Examples on geophysical and biomedical data are used to demonstrate the prowess of this high dimensional spectral representation.