How many complex species are there on Earth?

About 8.7 million.

And how many have we found to date?

About 1.2 million.

That is according to a recent study in PLoS Biology:

Knowing the number of species on Earth is one of the most basic yet elusive questions in science. Unfortunately, obtaining an accurate number is constrained by the fact that most species remain to be described and because indirect attempts to answer this question have been highly controversial. Here, we document that the taxonomic classification of species into higher taxonomic groups (from genera to phyla) follows a consistent pattern from which the total number of species in any taxonomic group can be predicted. Assessment of this pattern for all kingdoms of life on Earth predicts ~8.7 million (±1.3 million SE) species globally, of which ~2.2 million (±0.18 million SE) are marine. Our results suggest that some 86% of the species on Earth, and 91% in the ocean, still await description. Closing this knowledge gap will require a renewed interest in exploration and taxonomy, and a continuing effort to catalogue existing biodiversity data in publicly available databases.

Here is a table from the paper (click to enlarge) showing the vast difference in the numbers of catalogued and predicted species, which really highlights how little we know about life on our planet (particularly in the ocean).

An interesting calculation made in the study was the approximate cost and man power it would take to catalogue all of the species that are currently unaccounted for:

Considering current rates of description of eukaryote species in the last 20 years (i.e., 6,200 species per year; ±811 SD; Figure 3F–3J), the average number of new species described per taxonomist's career (i.e., 24.8 species, [30]) and the estimated average cost to describe animal species (i.e., US$48,500 per species [30]) and assuming that these values remain constant and are general among taxonomic groups, describing Earth's remaining species may take as long as 1,200 years and would require 303,000 taxonomists at an approximated cost of US$364 billion.

On the subject of species diversity, here is a nice 'tangled bush' image (click to enlarge then zoom in):