If central banks become shareholders, where can investors go? The ECB is buying corporate bonds. It could eventually also buy shares. What would be left for investors to invest in? Letting bank shareholders pay for bailouts resulted in crashes of their shares. It was interesting to see how fast they went down. It seems like bank shares are relatively volatile. Letting lenders take hair-cuts is very impopular (maybe this is a separate question) but, would the ECB becoming a shareholder / bondholder amount to the same thing? Could it soften or break the opposition against taking hair cuts? (Who would be paying ultimately?)
What would be left for investors to invest in?


Stocks and bonds, as always. Just imagine the CB as another institutional investor. Institutions invest in the capital markets all day every day. Another large player will put upward pressure on prices due to increased demand. Upward price pressure can either be good or bad depending on what side of the trade you sit. From the POV of other investors entering the market it could be bad. But for investors seeking to exit and the corporate management and employees who either own company stock or their compensation is impacted by their company's stock price, it could be good.


  ...would the ECB becoming a shareholder / bondholder amount to the same thing?


It depends. The scenario you describe simply amounts to a shift in the risk pool. The bond risk pool is shifted from sovereign debt to corporate debt. Which is inherently riskier but, again, it depends on which corporations. Risk quantification would require a thorough analysis in order to be useful.


  Could it soften or break the opposition against taking hair cuts?


Whose to say? Anything is possible. Further comment would be speculative.


  Who would be paying ultimately?


As with any bank or corporation, all profit and loss of the ECB is borne by its shareholders. In this case, these are the central banks of the EU's member states. Primarily, Germany, France and (soon-to-be formerly, post-Brexit) UK. Closely followed by Italy and Spain. See this page for more details.