Collinearity problem Consider a linear regression of this type: height: beta_0 + beta_1*weight.
Adding BMI as parameter would add complexity to the problem or just cause a collinearity problem?

Probably yes, we can suppose that BMI and weight are highly correlated. You can verify it by performing a correlation between BMI and weight and checking the resulting coefficient, but as in previous responses, they will correlate.
Accordingly, considering both in the same model could also heavily alter your outcomes (e.g. no predictor is significant). You should choose the most important for your research.
